Reaction-Diffusion Equation Solver
(Parabolic partial differential equations in two independent variables)
27 October, 2023

Copyright 1995-2023 Miroslav Kolá

V4.3-beta Release Notes

The rest is still for TRANSIENT V3.11
Except for the "packing list" at the end
All manual should be updated soon for V4.3

18 February, 2006

1. Introduction
2. Theory
    2.1. Time derivative discretization
    2.2. 3-point difference formulas for the spatial domain
    2.3. "Mass-balance" difference formulas
    2.4. Algorithm to solve the resulting algebraic equations
3. Vectorized algebraic-equations solver for TRANSIENT (NL3BANDV)
    3.1. Optional constraints (concentration bounds)
4. Conditions for terminating the iteration process in NL3BANDV
5. How to install and use TRANSIENT
    5.1. Installation
    5.2. Writing your own Problem Definition Module
    5.3. Running TRANSIENT-based programs
6. Overview of TRANSIENT
    6.1. Pseudocode for the TRANSIENT main function
    6.2. Adaptive time-stepping algorithm
    6.3. Precision control
    6.4. Exit codes
7. The details of writing your own PDM
    7.1. Header file with PDM-dependent input parameters
    7.2. Source code for the user supplied functions
8. Demonstration examples
    8.1 Demo 1. Uniform corrosion
    8.2 Demo 2. Gas generation and diffusion
    8.3 Demo 3. Stochastic surface growth
    8.4 Demo 4. Brusselator
Appendix: TRANSIENT V4.3-beta "packing list"

1. Introduction

Although it is often sufficient to know only the steady state of chemical systems, their time evolution (transient behavior) can be of interest, too. For example, when the transition into a steady state is very slow, or when the transient behavior itself is of the main interest, such as in corrosion or when a system oscillates about an unstable steady state. In such cases, numerical solution of the underlying reaction-diffusion equations is desirable. This solution can be conveniently obtained by finite difference methods. In this approach, all partial derivatives are replaced by suitable finite differences, and to propagate the solution in time, the resulting nonlinear algebraic equations are repeatedly solved numerically.

Spotnitz [1] introduced in 1994 a C package called NL3BAND implementing Newman's method [2-5] for solving banded nonlinear algebraic systems of equations, specifically those originating from electrochemical steady state problems. Newman's method is based on banded matrices and the Newton-Raphson (NR) method for solving systems of nonlinear equations.

Here I adopt this approach to the solution of arbitrary systems of coupled, nonlinear, parabolic partial differential equations in two independent variables (time and one spatial variable), and present a C package called TRANSIENT that can be easily applied to a large variety of such problems. TRANSIENT uses a general implicit differencing scheme for the time domain.

A list of all problem-independent modules constituting the TRANSIENT engine (solver) together with their brief descriptions is in the Appendix. The main (control) module of this package, transien.c, is based on an heuristic adaptive time-stepping algorithm that repeatedly invokes a vectorized version of NL3BAND (NL3BANDV), in which all matrices are treated as linear one-dimensional arrays. In this scheme, the magnitude of the next time step is determined also (or exclusively) by the number of NR iterations of the NL3BANDV solver.

For a given system of differential equations to be solved, the user must supply a problem definition (or description) module (PDM) consisting of the list of model input parameters (placed in a separate C header file) and a number of C functions (placed in one or possibly more C source files). Demonstration examples of such modules, demo1 to demo4 are supplied in subdirectories demo/demo1 to demo/demo4 of the TRANSIENT distribution package. These examples can serve as easy-to-follow templates for your own problems. Example No. 2 (demo2) also includes some TRANSIENT validation: it demonstrates a high degree of agreement of TRANSIENT's results with an analytical solution for this example. Example No. 1 (demo1) deals with a slow corrosion process reaching its steady state at about 4×108 a (years). Simulation of such processes with reasonable accuracy in relatively short CPU time seems to be the strength of TRANSIENT.

2. Theory

The purpose of TRANSIENT is to solve a set of n coupled partial differential equations of the parabolic type, e.g., of the form

on a finite domain x = (xleft, xright) for t > t0, starting from initial values ci(x, t0), subject to boundary conditions

k, i = 0, 1,..., n - 1; b =   left,right. Here x (position) and t (time) are the independent variables, ci are the n dependent (unknown) variables, k represents the equation number, and Rk are arbitrary nonlinear functions of all n unknowns ci and possibly also explicitly of position and time. In fact, TRANSIENT could handle a further generalization of eq. (1a) to equations of the type

where Gk are some nonlinear functions (see demo3.c). k, b can also be rather general nonlinear functions. Because in most TRANSIENT applications the dependent variables ci are expected to represent the concentrations of various species, ci are often called "concentrations" in what follows.

Additional constraints on ck may be set as described in Sec. 3.1.

To obtain numerical solution of eqs. (1), both x and t have to be discretized: there are N, generally nonequidistant, grid points xj; j = 0,..., N - 1, and an a priori unspecified number of time instants tl; l = 0, 1,.... Values of ck(x, t) will be calculated only at these discrete values of x and t; and ck(xj, tl) often abbreviated to ck(j, l ).

It is assumed that , and Rk in eq. (1a) are continuous functions of x and t, except perhaps at a small set of discrete values of x. Such discontinuities must be treated with care: for the highest accuracy use a spatial grid in which the position of each discontinuity coincides with a grid node, and use a consistent discretization. This will be discussed to some extent in Sec. 2.3 and also in Secs. 8.1 and 8.2 (demonstration examples demo1 and demo2).

All the derivatives in eq. (1) must be replaced by suitable difference formulas. This discretization is the responsibility of the PDM author. TRANSIENT is an open platform for the numerical solution of systems of eqs. (1), and must be provided with their discretized form. Difference formulas listed below represent some obvious choices. Most of them were used in the demonstration examples. My experience shows, that for the spatial domain, the mass-balance approach of Sec. 2.3 gives the best results.

2.1. Time derivative discretization

For the time derivative at the j-th grid point, one can always use

Here we assume that the solution is already known at tl and we want to advance it to the next time instant, tl+1.

2.2. 3-point difference formulas for the spatial domain

On the right-hand side of eq. (1a) the standard approach at points at which the discretized functions are continuous is to use the usual 3-point central difference formulas for the given type of grid in a differencing scheme of general implicitness 1 - , 01, mixing together the "old" solution at tl with the "new" (to be computed) solution at tl+1: is the fraction of the old solution in this mixture. Let us introduce

substitute this expression for all ci(x, t) on the right-hand side of eq. (1a), and then use the following central difference formulas for the derivatives by x at the j-th grid point:

where = xj - xj-1 and = xj+1 - xj. These formulas were obtained by requiring that the first three terms of the Taylor series at xj of the RHS of eqs. (4) and (5) are correct (i.e., equal to the respective LHS). If = = h, they turn into the standard equidistant formulas with the error term O(h2).

In evalf() of the demonstration PDM's 1, 2 and 4, (j)  cBar0[i], (j - 1)  cBarM[i] and (j + 1)  cBarP[i], in demo3.c it is hBar0, hBarM and hBarP.

Note that (j) as defined in eq. (3) correspond to ck(xj, t) at t = tl + (1 - )(tl+1 - tl) = tl + (1 - )tl+1 as calculated by linear interpolation between ck(j, l ) and ck(j, l + 1). Thus the last term in eq. (1a) should in this differencing scheme be replaced by Rk((j), xj, t). Similarly, if and depend explicitly on time, they should also be evaluated at t in the finite difference replacement of eq. (1a). In terms of variables exported from TRANSIENT into a PDM, t =  t - Tau / tStepInv. In demo1.c, t is stored in the variable tMean.

Usually, the best choice for should be = 0.5, corresponding to t = (tl + tl+1)/2. Note that only in this case the error of the difference formula in eq. (2) is of the second order in the time step, otherwise it is of the first order in time step. = 0.5 gives the Crank-Nicolson scheme, which for equidistant grids is of the second order both in space and time, and for pure diffusion is stable for arbitrary time-step size. It is expected to be stable for large time steps also for many types of the reaction term Rk(ci(x, t), x, t). = 1 gives the fully explicit scheme which is stable only for very small time steps, and = 0 gives a fully implicit scheme stable for arbitrary time steps. Thus, at first always use = 0.5, if there are any problems, try < 0.5.

The boundary conditions in eq. (1b) do not contain any time derivatives, and so they could always be discretized directly at t = tl+1. Thus, in k, left the following 3-point forward difference formula could be used at j = 0:

Here = x1 - x0, = x2 - x1, and f = /. Similarly, in k, right, any first derivatives at j = N - 1 can be written in terms of a 3-point backward difference formula

where = xN-1 - xN-2, = xN-2 - xN-3, and now f = /. However, in more complicated cases, better results (higher accuracy) will be obtained when consistently working at t at all nodes, and, if possible, using the approach discussed in Sec. 2.3.

2.3. "Mass-balance" difference formulas

When and are independent of x, using difference formulas (4) and (5) is equivalent to integrating eq. (1a) with respect to x (at t = t) between two neighbouring grid midpoints xj - /2 and xj + /2, and then using the obvious 2-point difference formulas for the first derivatives and for the function values at the midpoints, i.e.

    and similarly for the xj + /2 midpoint.

To obtain a consistent discretization throughout the model, it is desirable to extend this integrational or "mass-balance" approach to all the grid nodes, including the model end-points and the nodes with discontinuities. For the model end-points, the integration is done from xleft to xleft + /2 (or from xright - /2 to xright). One will still use eqs. of type (7) at the first (or last) grid midpoint. But at xleft (or xright) one will use the solution of eqs. (1b) for the unknowns (xb, t) obtained in terms of ci(xb, t) and possibly t. For example, for the trivial case of von Neumann b.c., = , this solution is simply equal to Const. In case of nonlinear form of eq. (1b), one has to solve it numerically at the beginning of each NL3BANDV (see Sec. 2.4 and Sec. 3) iteration. This process will be discussed in more detail in Sec. 8.1 (e.g., eq. (32)).

For the inter-media interfaces, i.e., at grid nodes with discontinuities in , and R in the interior of (xleft, xright), the integration of eq. (1a) in the vicinity of such a grid point must be performed separately in (xj - /2, xj) and in (xj, xj + /2). At both outer ends of this interval, one can still use eqs. (7). At xj then the discontinuity relations between ck (or ck/x) on two sides of xj (resulting e.g. from the continuity of the diffusion flux at xj). More details can be found in the demonstration examples of Secs. 8.1 (e.g., eq. (31)), 8.2 and 8.4.

This approach is called "mass-balance" discretization because it divides the whole model into "integration" elements extending from one grid midpoint to the neighbouring one (or to the end-point of the model). It conserves mass locally everywhere by accounting for the mass fluxes crossing the boundaries between these elements or through the model faces. It is actually a convection-less equivalent of the Nodal Point Integration method of Runchal [6] used for two- and three-dimensional spatial domains. Runchal [6] also reports that his method gives high precision results. It is a simpler relative of the finite element method (or a hybrid between the finite element and finite difference methods).

2.4. Algorithm to solve the resulting algebraic equations

Upon discretizing eqs. (1) as described above, we get a system of n×N equations of the form

in n×N unknowns ci(m, l + 1), involving the "old" concentrations, ci(m, l ), as known parameters. Here j = 0, 1,..., N - 1; m = 0, 1, 2 for j = 0; m = j - 1, j, j + 1 for 1jN - 2; and m = N - 3, N - 2, N - 1 for j = N - 1. For 1jN - 2, the form of Fk(j) evidently is

For j = 0, N - 1 it may differ depending on the exact form of boundary conditions (1b).

The set of, possibly nonlinear, algebraic equations (8a) is solved by the NR method of ref. [1]. In this approach, Fk(j) are expanded about some trial point, indicated by , and only linear terms of the Taylor series expansion are kept: e.g., Fk(j)(ci(m, l + 1), ci(m, l )) is, for 1jN - 2, approximated by


Note that Ak, i(j) are the components of an n×n matrix defined for j = 1 to N - 1, Bk, i(j) of an n×n matrix defined for j = 0 to N - 1, and Dk, i(j) of an n×n matrix defined for j = 0 to N - 2.

Do not forget the factor 1 - = (j)/ck(j, l + 1) (denoted New in the demonstration PDM's) in the contributions to the Jacobian submatrices (10), (11) and (12) for j = 1 to N - 2 from the first two terms (generalized diffusion term and the reaction term Rk) of Fk(j) in eq. (8b).

Because 3-point difference formulas (6) may be used at the end-points, the equivalents of truncated expansion (9) for j = 0 and N - 1 contain two more matrices with possibly nonzero elements: with components

and with components

Substituting all these truncated expansions for Fk(j) into eq. (8a), one gets a linearized set of equations that in matrix form reads


Here and are vectors with n components ci(j, l + 1) - ci(j, l + 1) and Fk(j), respectively.

The solution strategy is to evaluate and at some = , and solve eq. (15) for . If the absolute value of (and preferably also ) approaches zero, the vector = + is accepted as the solution. Else is used as the new trial point and the process is repeated until convergence is obtained, or signs of divergence or oscillations are detected. In our case, the most suitable initial guess for the unknown ck(j, l + 1) is obtained by linear extrapolation of ck(j, l - 1) and ck(j, l ) to the next time instant tl+1.

In order to solve eq. (15), a memory-use-minimizing modification of the procedure of ref. [1] is used: First the upper-lower decomposition of J is done to get

and can be written as


so that

Now we can calculate the entries of the and matrices simultaneously with the components of the vector by re-using the same memory for the Jacobian submatrices (10) through (14) and also for and for all grid nodes j (re-using the same three n×n matrices , , ; note that and are never used simultaneously, the same is true for and ). Similarly, a single n-component vector is used for all . Only and - have to be stored simultaneously for all j (N×n×n elements of , and N×n elements of xi).

The algorithm is as follows:

   for j = 0 do:                  
(0)     -

   for j from 1 to N - 2 do:  
- (j - 1)  
- (j - 1)    
if (j = 1) { - }  
(j)     -

   for j = N - 1 do:            
- (N - 3)    
- (N - 2)    
- (N - 3)  
- (N - 2)  
(N - 1)     -

In the end, the back substitution of eq. (17) is used to solve for the - , which will again be stored in (j):

 for j from N - 2 to 1 by -1 do (j) - (j(j + 1) (j)     -
 for j = 0 do: (0) - (0) (1) (0)  
  (0) -  (2) (0)     -

At this point, - is stored in (j) for all j = 0,..., N - 1, thus = - .

3. Vectorized algebraic-equations solver for TRANSIENT (NL3BANDV)

In the TRANSIENT code, the number of equations n is stored in the variable neqns, and the number of nodes N is in nnodes in the file transien.c and in NJ in nl3bandX.c (X stands for v or 1).

The algorithm described in Sec. 2.4 is implemented in the function runband() in file nl3bandv.c. For matrix inversion and LU decomposition, modules matinvv.c and lufacv.c are further needed. A simplified version of nl3bandv.c for = 1 is in the file nl3band1.c. This version does not need any other modules, and is almost two times faster than nl3bandv.c called with = 1 (compare e.g. the speed of demo2a and demo2av of the standard demonstration set; see Sec. 5.1.). The names and arguments of functions defined in nl3bandv.c and nl3band1.c are identical.

nl3bandX.c repeatedly invoke evalf() to calculate and matrices (10) through (14) for one j at a time. evalf() must be supplied by the user as part of a PDM.

Because the dimensions (n and N) of the various two- (three-)dimensional arrays involved will vary from problem to problem, these arrays are allocated memory dynamically. To be able to refer to them in all modules as to multi-dimensional arrays, they would have to be declared as pointers to array of pointers (to array of pointers). This would consume some memory and may slow down the execution because of a series of pointer evaluations for each matrix element. This was not an issue when a single set of equations was solved per problem in the original steady-state-only NL3BAND of ref. [1]. TRANSIENT may handle hundreds of thousands of time steps in a single run, i.e., hundreds of thousands of runband() invocations. Then any savings in CPU time may be important. Therefore, TRANSIENT uses a "vectorized" version of NL3BAND, NL3BANDV, that stores all multi-dimensional arrays as linear vectors ("row by row", i.e., the last subscript varies the fastest). These vectorized modules have file names that end with a "v" ( ... v.c).

In a PDM (see section 5), the number of equations (row dimension of all matrices involved) can always be declared a constant, e.g.,
    #define NEQNS  neqns
Then one can define a new type - vector of length NEQNS, e.g.,
    typedef double ROW[NEQNS]
and treat all the two-dimensional matrices as pointers to ROW (see demo1.c and demo4.c). Thus, when writing a PDM, one can still enjoy the convenience of the normal matrix notation when dealing with concentrations or the Jacobian submatrices (10) to (14), and at the same time save some memory and especially the CPU time. To this end, the order of matrix elements in TRANSIENT's dynamical vectorized storage of matrices is the same as in matrices declared with fixed numbers of elements. Just apply the type cast operator when passing pointers to matrices between transien.c and a PDM: (ROW*) when passing a pointer from TRANSIENT to a PDM, and (double*) in the other direction.

3.1. Optional constraints (concentration bounds)

Bounds on the range of each dependent variable (concentration) may be set up to be enforced in runband after each iteration. This is done with the help of bound definition arrays (BDA) described below. The addresses of two arrays of pointers *LB[NEQNS] and *UB[NEQNS] must be supplied from a PDM through the arguments ***LBp and ***UBp of the function initNL3BANDV(). LB[k] points to a BDA that defines the lower bound for the k-th dependent variable. Similarly, UB[k] points to a BDA that defines the upper bound for the same variable. If the bounds for several different variables are the same, only one BDA needs to be set up for all of them, with all the respective LB or UB pointers pointed to it. If there is no upper or lower bound in effect for the k-th concentration, UB[k] or LB[k] may be set to NULL. Similarly, if nothing is assigned to **UBp or **LBp (predefined to NULL), no upper or lower bounds are set for all dependent variables. More details are at the beginning of nl3bandv.c.

Individual bounds for the k-th dependent variable may be changed during a transient calculation by calling the function changeBound(int k, char which, double *BDA), where which is either 'l' or 'u' for the lower or upper bound, respectively, and BDA points to the new (modified) bound definition array.

BDA are designed so as to minimize the memory and CPU time requirements for the bound checking. Four types of bounds are recognized, distinguished by the value of the 0-th element of a BDA:
    0 - no bound is defined (no further BDA elements are required); this is equivalent to setting UB[k] or LB[k] to NULL;
    1 - a constant bound (which is equal to the next, i.e., the 1st, BDA element) applies for all nodes;
    2 - defines a piece-wise constant bound (an arbitrary even number of BDA elements must follow, each couple consisting of a bound value followed by the index of the last node for which this bound holds; the last couple is recognized by the value of its node index which must be either smaller than the previous node index (e.g. equal to 0), or be N - 1);
    3 - sets a different bound for each node (must be followed by N values of bounds for nodes 0 through N - 1).
    4 - can be used for lower bound only to suppress the numerical noise; sets the same zero cutoff Zc for all nodes; Zc is equal to the next (1st) BDA element. Usage: if ci < Zc, ci is set exactly to 0.

The function startband() checks whether the bounds are defined consistently - whether, if both defined for the same k, the lower bounds are less than or equal to the corresponding upper bounds at all nodes.

4. Conditions for terminating the iteration process in NL3BANDV

TRANSIENT V3.11 is designed to minimize the number of situations when a transient solution run would fail to complete (i.e., fail to reach an arbitrary final time tEnd). The run is not being terminated if a certain predetermined level of NR precision cannot be achieved. It reports to the user the achieved precision as indicated by the quantities cresidT through fresidM defined below. This is done periodically in the headers of the profiles output files, and with monitor=2 also continually on the screen.

Currently only the following value is used in the convergence criteria of the NR iterative algorithm described in section 2:

For the error monitoring purposes, also the following values are calculated:

and if monitor=2 is set, also

Basically, NR algorithm continues as long as cresidT decreases significantly with respect to its value in the first NR iteration averaged over time from the beginning of the run. The degree of this decrease can also controlled by the the control input parameter Prec (precision). The default value of Prec=0 corresponds to a decrease by a factor of 10-4 (if achievable), a general value to a decrease by 10-(4+Prec) (the larger Prec, the better convergence is required; small negative values of Prec are allowed and give smaller than default precision, positive Prec larger than default precision). It is expected that the degree of convergence defined by Prec is achieved within nIter NR iterations. TRANSIENT is trying to adjust the magnitude of the time step to the value of nIter. For the details of these convergence conditions see the file nl3_conv.h (which can be easily customized by the user if necessary), and also the line converting the value of the variable Prec in nl3bandX.c itself.

The "concentration" bounds are enforced immediately after each NR iteration only if the macro NR_BOUNDS is set to 1 (see defs.h).

5. How to install and use TRANSIENT

5.1. Installation

Create an empty directory on your hard disk that will serve as the TRANSIENT root directory. Let us call it TRD (in what follows, all the directory and file names will be given relatively to TRD, unless they explicitly contain TRD; except that sometimes no path is given when it is absolutely obvious. If you got TRANSIENT on a CD, just copy the contents of the CD into TRD.

The preferred way is to get TRANSIENT from the Web. Download transientVx.xx.tgz (not available right now, sorry),
and unpack this archive into the TRD:

Then read the file demo/0readme.txt containing instructions on how to make a standard set of demonstration programs, and how to maintain TRANSIENT objects for linking with your own PDM. (Quick guide: on a Unix-like system: type 'make' in TRD, and follow the instructions; similarly in all demo/demoX directories; for MS C compilers check makefile.VisualCpp, or the OBJECTS and demo\demoX\DEMOX makefiles; on VMS, use MAKEDEMOS.COM.)

The exact configuration of TRANSIENT (inclusion of various optional features) is controlled by the current state of the file defs.h, which can be edited to suit a particular PDM before the compilation and linking of the TRANSIENT modules with a PDM. The settings in this file are fully respected by the Unix and MS makefiles residing in the individual demo/demoX directories.

However, when the standard set of demonstration programs (for the four demonstration PDM supplied with TRANSIENT) is made by the Unix or Visual C++ makefiles in TRD or by the VMS MAKEDEMOS.COM command procedure, the current state of defs.h is overridden by the compiler command line flags (defs.h is written so that command line flags have precedence). The standard set represents a selection of possible TRANSIENT configurations as used with the sample PDM supplied with TRANSIENT. Although the standard set is made without changing the state of defs.h, it represents partly what can be set in defs.h.

The standard set will appear in directory demo/bin, and consists of these seven programs:

demo1, demo2a, demo2b, demo3, demo4
- executables for all demonstration PDM corresponding to the "default" (i.e., as found in transientV3.11.tgz) state of defs.h.
- demo1 with PDM function tracing enabled. (defs.h: #define PDMtrace  1)
- the same functionality as demo2a, but linked with nl3bandv.c instead of the nl3band1.c used by demo2a (comparing the CPU usage by demo2a and demo2av compares the speed of the two variants of NL3BANDV available for neqns=1). (Cannot be set in defs.h; must be done in the linking stage.)

5.2. Writing your own Problem Definition Module

To use TRANSIENT for their own problems, the user must prepare a PDM consisting of a C header file containing the list of all the model parameters, and of one or more C source files with the code for the following functions:

int get_neqns_etc(void)
void generate_grid(void)
void Initialize(double *cp, double *c_op)
void initNL3BANDV(double***LBp, double***UBp, double*F, double*A, double*B, double*D)
void auxiliary_parameters_for_evalf(void)
void evalf(int j)
void pr_profiles(int MaxIter)
void read_profiles(char *filename)
void init_Other(void)
int update_Other(void)
void pr_Other(int only_Other)

More details will be given in Sec. 7. Use the source, header and make files from the individual demo directories as templates for writing and maintaining your own projects (PDM). After compilation, link the PDM together with the modules that constitute TRANSIENT, contained in TRD.

5.3. Running TRANSIENT-based programs

All the input and output is handled by the TRANSIENT engine, and is thus essentially the same for all PDM. Let us use demo1 to illustrate how to run any TRANSIENT-based code. Only the first three lines below are demo1 specific. For other PDM, they would be replaced by the title specified in get_neqns_etc() (see Sec. 7), or by the string "Unnamed PDM". Also, any occurrence of 'demo1' on the second help page would be replaced by the name of the respective executable.

Typing just demo1 without any command-line options (may have to be preceded by its path, such as ..../TRD/demo/bin/demo1) will produce this help page:

                   Demonstration No. 1
  Uniform Corrosion and Transport of Corrosion Products
  .-----       p o w e r e d      b y      -----.
  | TRANSIENT V3.01 nonlin parabolic diff equation solver |
  `-----     -----'
 To get help on how to run this program, enter:  ???
 If you continue now, default output files _T_*, par_T_, diag_T_ and
 stat_T_ will be created.
 If you end input now, default input parameter values would be used.
 You can first modify any input parameters in arbitrary order, one at
 a time, by entering the name of an input parameter followed by its
 new value, separated from the name by spaces or/and an equal sign.
 A missing value is interpreted as a zero number, empty character or string).
 To end input (start calculation), enter  end  or  press  Ctrl+D .
 At any time during this input phase, you can also enter:
  ?c-      to list the current values of all control parameters;
  ??c-     to also get extensive help on the role of each control parameter;
  ?m-      to list the current values of all model parameters;
  ??m-     to also get any available help on model parameters;
  ?<str>   to list the current values of all input parameters whose
                names start with <str>;  e.g. ?tSt;
  ??<str>  to also get available help for input parameters starting with <str>;
  ?        to reprint this interactive menu;
  ex[it]   to terminate execution;
  qu[it]         - " -
  by[e]          - " -
  Ctrl+C   will terminate execution at any stage.

This was produced in Linux. In MS Windows or VMS, Ctrl+D would be replaced by Ctrl+Z.
Entering now '???' will produce:

                    Demonstration No. 1
   Uniform Corrosion and Transport of Corrosion Products
  .-----       p o w e r e d      b y      -----.
  | TRANSIENT V3.01 nonlin parabolic diff equation solver |
  `-----     -----'
 To get help:  demo1 -h  ...  prints this page
               demo1 -d  ...  lists all input parameters and
                                     their default values
               demo1 -p  ...  as -d plus any available info on parameters
               demo1 -c  ...  lists only control parameters,
                                     their purpose and default values
 To run:  demo1 ['Output' [[<] 'Input']]
     'Output' ... "root" of names of output files: if SepProf=0,
        there is a single file 'Output' for (concentration) profiles;
        otherwise profiles for I-th output time are written into 'Output'I;
        Input parameters and all the optional output generated by
        init_Other, update_Other and pr_Other are written into a
        file whose name is created by adding prefix "par" to 'Output'.
        Diagnostic output (minima and maxima of tStep, cDiffM, and
        all iter between pr_Other times, or at most over 1000 time
        steps) is written into a file with prefix "diag" added to 'Output'.
        CPU times and more are written into a file with prefix "stat".
     'Input'  ... name of the input file; only some or none input
        parameters need be read in, defaults are used for the rest.
     Default files: 'Input':       `stdin`
                    'Output':      _T_*
                     par'Output':  par_T_
                     diag'Output': diag_T_
                     stat'Output': stat_T_

and the execution will terminate. This second help screen is identical with what one gets after typing demo1 -h .

For example, to run TRANSIENT (demo1) with the default values of input parameters, or to change some of the input parameters interactively at the beginning of the run, and store the results in files parD1o, diagD1o, statD1o and D1o (or D1o1, D1o2, ..., depending on the final value of the input parameter sepProf), enter 'demo1 D1o'. When finished with modifying input parameters, entering 'end', or pressing Ctrl+D (UNIX/Linux/Cygwin), or entering Ctrl+Z (MS native), or pressing Ctrl+Z (VMS) will start the calculation proper.

Alternatively, one can prepare an input file containing all input parameters the values of which differ from the defaults hard-coded in TRANSIENT and in a PDM. The same rules as for the interactive input apply: one parameter per line; order of parameters is not important; in case of duplicate entries, the last one is effective; for parameters not present in the input file, default values are used. Each line consists of a parameter name followed by its value, separated by space or/and an equal sign. The full syntax of an input line is:
[comment char] [w] name [w] { [=][w] | [{=|w} [w] value [w [anything]]] }
Here 'w' represents any sequence of white spaces, [ ] delimit optional items, and { | } alternative items. The only mandatory item is the parameter name. A missing value is interpreted as a zero, null character or empty string.

Examples of input data files are in all demo/demoX subdirectories (data1, dataS1, data2, etc.). To use e.g. data1, specify it after the output file root name: such as 'demo1 D1o data1'.

The first character of each input line can be the comment character CmntC (see below). This feature allows to use directly a parameter output file (parOut), perhaps after some editing, as a new input data file. The comment character can also change inside an input file. In this way, whole groups of lines in the input file may be "switched" on or off between runs by editing a single line defining CmntC (see e.g. '#' and '%' in demo/demo1/data1) - the lines starting with a character that is not the current CmntC will be ignored, and the "Unknown input parameter name" warning issued.

To capture the list of all input parameters, their default values and description into a file, type
'demo1 -p >& file_name' (UNIX) or 'demo1 -p > file_name' (MSDOS). The file ends with whatever information (names, values, and description of their purpose) on the problem-dependent input parameters the user supplies in the PDM header file (in this case in demo1.h).

Description of all control input parameters can be found in the file controls.txt. It was generated by demo1 -c >& controls.txt. This list would be the same for all PDM, except that the default values of those control parameters that are exported from TRANSIENT to a PDM (see transien.h) may differ, as they can be reset in the PDM to suit better the modeled system.

If some of the right-hand sides of eqs. (1a) or of the left-hand sides of eqs. (1b) depend explicitly on time, then steady state has no meaning. However, in such a case, steady >= 0 could be used for example to calculate a possible final state corresponding to t = . This could be achieved by setting all the terms in evalf that depend explicitly on t to their limit values corresponding to t = when steady >= 0.

6. Overview of TRANSIENT

The relation of some of the variables used in TRANSIENT (transien.c) to the notation of section 2 is as follows: tStart  = t0, t  = tl+1, tt  = | tl+1 - t0|, ttOld  = | tl - t0|, Tau  = , tStep  = tl+1 - tl, ttStep  =  |tStep|, tStepInv  = 1/(tl+1 - tl), tStepInit is the initial value of ttStep, tStepMax is the largest allowed value of ttStep.

6.1. Pseudocode for the TRANSIENT V3.11 main function

Function main is located in module transien.c. The user supplied PDM functions are shown in red.

void main(int argc, char **argv) {

< <Start CPU clocking; set up signal catching> >
neqns = get_neqns_etc();
read_input(argc, argv);
if(steady) Tau = 0.;
open_output(argc, argv); /* Generates headers of all output files */
pr_par(-1); /* Writes out all input parameter values into the par'Output' file */
< <Allocate memory for c, c_o> >
Initialize(c, c_o);
if(*init_from_file != 0) read_profiles(init_from_file);
startband(neqns,nnodes,c); /* calls initNL3BANDV(...) */
if(steady) {
        tStepInv = 0.;
        c_o = c; /* Only in TRANSIENT, not in a PDM */
        attempt = 1;
        /* runband invokes repeatedly evalf(node) */
        if(runband(0.0) < 0) {
            if(attempt < attemptMaxSteady) {
                if(monitor) < <Report on the current attempt> >
                Randomize(10*attempt+iter, 1e-3*max(| ck(j, l )|), c, c);
                attempt+ +;
                goto Try_steady_again;
            < <Report FAILURE after attemptMaxSteady(=1000) attempts> >
        if(steady = = 1) exitTR(0);
        else { /* steady==2 */
            < <Restore Tau and c_o> >
            auxiliary_parameters_for_evalf(); // 2nd time for steady=2
        } /* End of: if(steady) */
/* Transient calculation: */
< <Set tSign = sign(tEnd-tStart) ;
    transform the real time t to an always nonnegative tt = |t - tStart|,
    i.e., ttEnd = |tEnd - tStart|, tOutInit = |tOutInit - tStart| > >
ttOld = ttOut = 0.;
tt = ttStep = tStepInit;
DttOut = tOutInit/nOutOTH;
highAttempts = 5;
< <etc.> >
init_Other() // 2nd time for steady=2;
< <Clock the beginning of the transient calculation; for stat'Output'> >
while(ttOld < ttEnd) {
    for(k=1; k<=nOutOTH; k+ +) {
        ttOut += DttOut;
        if(ttOut >= ttEnd) { ttOut = ttEnd; k = nOutOTH; }
        if(tt > tOut) { tt = ttOut; ttStep = ttOut - ttOld; }
                            /* This can happen when ttStep>DttOut */
        do {
                t = tSign * tt + tStart;
                tStep = tSign * ttStep;
                tStepInv = 1./tStep;
                /* runband invokes repeatedly evalf(node) */
                if(runband(t) < 0) {
                    if(ttStep > tStepMin) {
                        if(monitor>1) < <Report on the current attempt> >
                        attempt+ +;
                        < <Decrease ttStep, adjust tt, and reset c[j][k] > >
                        goto Try_again;
                    // Can happen only rarely, for singular matrices:
                    < <Report FAILURE> >
                cDiffM = | ck(j, l + 1) - ck(j, l )|
#if !NO_EXIT_m4
                if(cDiffM = = 0.0) exitTR(-4); /* A stable steady state was reached */
                if(strict> =-1) Scale = cDifMxf(tt) / cDiffM; /* cDifMxf(tt) is determined by input
                                parameters cDifMxMode, cDifMxInit, cDifMxEnd */
                if(strict= =1 && Scale<1. && ttStep>tStepMin) {
                    if(monitor>1) < <Report on the current attempt> >
                    < <Reduce current ttStep; interpolate c> >
                    attempt+ +;
                    goto Try_again;
                if(attempt>1) highAttempts += attempt;
                                    else if(highAttempts>1) highAttempts- -;
                < <Adaptive time-stepping algorithm:
                        Determine the new time step using some or all of:
                        strict, Scale, highAttempts, magnitude of the last two steps.
                   Adjust the new time step if:
                        ttStep>tt (to limit time step initially)
                        or ttStep>tStepMax
                        or ttStep>ttOut and ttOld<ttOut.
                > >
                if(monitor) {
                    < <Print t, cDiffM> >
                    if(monitor>1) < <Print tStep, Scale> > else < <Print attempt, iter> >
                    < <Print # of time steps, of calls to runband(), and of all iterations since
                            the beginning of the run> >
                < <Extrapolate to get initial guess for c[j][k] for new time> >
                < <Append diag'Output' file every now and then> >
                tStep = tSign * ttStep;
                i = update_Other();
                if(i) out_Other(...); /* calls pr_Other(...) */
                if(i<0) exitTR(1000-i);
#if RESETtStepAT99
                if(i>1) < <Reset ttStep according to the value of i> >
                    if(tSign*tStep < ttStep) { /* tStep reduced in update_Other */
                        ttStep = tSign*tStep;
                        tt = ttOld + ttStep;
        } while(ttOld < ttOut); /* End of: do */
                out_Other(k<nOutOTH); /* calls pr_Other(k<nOutOTH) */
        if(monitor= =0) < <Print summary information> >
    } /* End of: for(k=1; k<=nOutOTH; k+ +) */
    out_profiles(tt<ttEnd); /* calls pr_profiles(maxIterDone) */
    if(tOutMulF > 1.) DttOut *= tOutMulF;
} /* End of: while(ttOld<ttEnd) */
} /* End of: main() */

6.2. Adaptive time-stepping algorithm

An heuristic adaptive time-stepping algorithm (ATSA) is implemented. When strict  < 0, the time step is determined solely by nIter as described in Sec. 4. For all values of strict, the algorithm tries to choose the largest time step such that at most nIter NR iterations are needed in runband.

The current ATSA is based on the lengths of two previous time steps. From their ratio, and from whether the current number of iterations exceeds nIter or not, ATSA tries to determine a suitable next time step. This extrapolation is always biased towards larger time steps (it tries to push the step upwards whenever possible). If for the current time instant, NL3BANDV detects a really bad NR convergence, the time step is reduced, and another attempt at convergence is made. Too many recent repeated attempts signal that the time step has been increased too fast. The variable highAttempts keeps track of this (see Sec. 6.1). The larger its value is, the smaller is the upper limit on the maximum allowed time step increase. For further details, inspect the file transien.c.

With strict  0, in addition to what is described in the previous paragraph, ATSA also compares cDiffM, which is the maximum change in ck(x, t) over the current time step, with the maximum allowed by the user, specified through the input parameters cDifMxMode, cDifMxInit, cDifMxEnd that determine the form of the cDifMxf(tt) function (see Sec. 6.3). If strict  = 1, the algorithm makes sure that cDiffMcDifMxf(tt) almost "strictly".If strict  = 0, cDiffM may exceed cDifMxf(tt) somewhat. The larger the cDifMxf(tt), the larger the time steps (provided that nIter allows that) and the shorter the total execution time, but also the larger the expected error.

One can watch the ATSA action with monitor 2.

6.3. Precision control

The NL3BANDV solver of the discretized nonlinear equations always tries to achieve the highest precision it can for a given time step, as described in Sec. 4. The values of strict, nIter and cDifMxf(tt) control the error caused by the discretization of time. Number and spacing of the x-grid nodes control the error introduced by spatial discretization.

The relaxed version of the ATSA algorithm corresponding to strict  < 0 will essentially find the largest cDiffM (as a function of time) compatible with a given value of nIter. Often, this may already give very good results with a suitably chosen value of nIter.

Thus a recommended strategy for TRANSIENT V3.11 is to start with strict  < 0 and a sufficiently high value of nIter (at least 10). If the resulting cDiffM is too large (check the contents of the diag'Output' file), or a fatal failure with the exit code -8 is encountered, first try to decrease the value of nIter. If that alone will not help, use strict    0 and suitable values of the parameters of cDifMxf() (initially again with a higher value of nIter). In principle, one should repeat this process (decrease the values returned by cDifMxf() further, and/or decrease the value of nIter) until the results for concentrations do not change any more. Of course, using very small values of cDifMxf(tt) with strict    0 can increase the execution time considerably. The same may be true if using very small values of nIter.

cDifMxf() is actually a pointer to a function. This pointer is set at the end of read_input (in InOut.c) to point to a suitable function according to the value of cDifMxMode. To run a TRANSIENT-based program, it is enough to know that cDifMxf() returns the following values:

A. cDifMxMode  = 0:

B. cDifMxMode  = 1:

(linear interpolation)

C. cDifMxMode  = 2:

(log-log linear interpolation equivalent to =   ;
can be used only if tStart and tEnd have the same signs)
Note that for the same values of cDifMxInit and cDifMxEnd, case C usually produces much smaller cDifMxf(tt) values than case B.

6.4. Exit codes

On termination, TRANSIENT returns the following status code:
    0 : on successful termination;
    -4 : when a steady state is reached in transient calculation (rare;
            can be switched off by nozero NO_EXIT_m4 - see defs.h);
    -3,-2,-1 : on improper initializations during TRANSIENT's startup;
    -5 : when unsupported type of a model input parameter used in PDM;
    -6 : when number of grid nodes changed after steady state calculation;
    -7 : when number of grid nodes increased above the initial value;
    -8 : when singularity encountered repeatedly in NL3BANDV;
    > 100 : when a signal is caught (status = signal + 100);
    1 ... 100 : can be set in a PDM using exitTR(status) to flag various termination conditions.
    1000+ | i|, where i < 0 is a negative return value of update_Other.

These exit codes will appear at the end of the stat'Output' file. When you have to terminate TRANSIENT from a PDM, use exitTR(status) instead of exit() (see demo/demo1/demo1.c for an example). exitTR will update all the output files with the results available at the time of termination, complete the last line of the diag'Output' (diag_out) file, and generate the summary in the statistics file, stat'Output' (stat_out). Use exit code 1  status  100, or 200  status  1000.

The above codes are passed through the C exit function, which accepts any integer as its argument. The contents of the shell exit status variable then equals this C exit status modulo 256. E.g., in the Unix csh shell, 'echo $status' returns values from -128 to 128, but in sh or bash shells, 'echo $?' returns values from 0 to 255. Thus sh and bash will change the TRANSIENT's internal exit statuses -8, -7, ..., -1 into 248, 249, ..., 255, respectively (this has of course no effect on the termination codes and messages that may be printed at the end of the stat'Output' file).

7. The details of writing your own PDM

PDM has two parts described in Secs. 7.1 and 7.2.

7.1. Header file with PDM-dependent input parameters

All information about the PDM-dependent input parameters is contained in a separate C header file. This file will be inserted (#include) into transien.h. In turn, transien.h is #included into the PDM source file(s). In this way the control and PDM-dependent parameters are integrated and handled in the same way. transien.h will extract from the PDM header file all the information needed for the declaration, input, output, and possible allowed-range checking of the PDM-dependent input parameters. Placing them into the PDM header file makes them global variables in the PDM source file(s), provided that the format of the beginning of source file(s) conforms to Sec. 7.2. They must not be declared again in the PDM source file(s).

The PDM header file may only contain arbitrary number of IN_PAR, IN_STRPAR and CHECK_PAR macros and C comments:

is used for scalar numerical input parameters, and has the following format:
IN_PAR(name, type, default, format, description)
The arguments are:
- the name of the input parameter (a valid C identifier) as used in the code of the PDM source.
- the type of name (a valid C type): double, int, long, char, ...
- the default initial value of name (a valid C constant) - to be used if no value for name is read in (from an input file, or interactively).
- a valid C format string for name; may include e.g. the units of name; the opening and closing double quotes used to delimit a string must be omitted; example: %+.12g sec
- an optional arbitrary long string (can use newlines \n, tabs \t, spaces \040, etc. for formatting) describing the role and/or the range of values of name; will be printed out in response to ??name or ??m- during the interactive input phase, or will appear in the output generated with the -p command-line flag (see Sec. 5.3). description must not contain any commas (except inside quotes, or entered using their octal ASCII code as \054), or unclosed single or double quotes (when not entered as \047 or \042, respectively).

is used for string input parameters. Possible use: to have switches with mnemonic string values between various model features, or to be able to vary the output format for concentration profiles (a few significant digits are enough for plotting, but more digits may be required when the profiles are to be used as the starting point for another run - see input parameter frmt in demo1). The format is:
IN_STRPAR(name, length, default, description)
The arguments:
name, default and description
have the same role as in IN_PAR, and default is now a valid C string constant which must be enclosed in double quotes.
is used in the declaration of name: among others, IN_STRPAR macro is converted to:
    char name[length] = default;
length must be large enough to accommodate the longest string intended to be the value of name plus the terminating '\0' character.

was originally designed to define the range of legal values of numerical input parameters to be checked at the end of the input phase. It has a single argument - a valid C Boolean expression that can involve any combination of all (PDM-dependent and control) input parameters. The execution will be terminated if this expression is not true for the values entered. No CHECK_PAR line need be present. The CHECK_PAR macros may also be placed anywhere in a PDM, and may test a Boolean expression involving any variables, not only the input parameters.

The PDM header file could be completely empty in the rare case that no model parameters will be changed between the runs. But it must always be present. Let us conclude this section with some examples of the entries of a PDM header file:

IN_PAR(L1,double,0.5,%g m,Thickness of medium 1)
IN_PAR(interv2,int,45,%d,Number of intervals in "[L1,L1+L2]")
IN_STRPAR(frmt,10,"%.8g",format for double for concentrations & corr current & potential)
/* A space will be added at the beginning -> max usable length = 8: */
/* description may be missing: */
IN_PAR(A,double,1.5e4,%g mol/cm 2,)
IN_STRPAR(BCr,8,"massBal",Type of formula for the zero flux RHS b.c.: \n \040 \040 \
        massBal - full mass-balance equation for the last half interval; \n \040 \040 \
        3point - standard three-point discretization formula)
CHECK_PAR(!strcmp(BCr,"massBal") || !strcmp(BCr,"3point"))

7.2. Source code for the user supplied functions

The list of the functions that must be defined in a PDM was given in Sec. 5.2. They may be all placed in a single source file, or may be distributed among several files. If there is only one such source file, it must have the following two directives before all declarations:
#define PDM_H "PDMpath/PDMheader.h"
#include "TRDpath/transien.h"

If a PDM consists of more than one file, one of them (it does not matter which one) must start again exactly as above. In all the other ("additional") source files, an additional #define directive must precede the above two lines. These additional source files must start as follows:
#define PDM_H "PDMpath/PDMheader.h"
#include "TRDpath/transien.h"

Here `PDMheader.h' is the name you gave to the PDM header file described in Sec. 7.1. `PDMpath' is its path relative to TRD, and `TRDpath' is the path of TRD relative to the directory where the PDM source file will be residing. Or one can use absolute paths. Everything else in the above two (or three) lines is literal, including the double quotes. PDM_H is a "place holder" for the name of the PDM header file in the transien.h code. For example, for demo1, the above directives are:

#define PDM_H "demo/demo1/demo1.h"
#include "../../transien.h"

Template for PDM source files is in the file PDMtemplate.c. In the rest of this section, one will find further details on how to write PDM source code.

All six functions get_neqns_etc, generate_grid, Initialize, read_profiles, auxiliary_parameters_for_evalf and initNL3BANDV are essentially initialization routines. They are invoked in TRANSIENT in the above order (see Sec. 6.1; read_profiles is called only if the init_from_file input parameter is assigned the name of an existing-file). All necessary initializations may be divided among them more or less arbitrarily, except that get_neqns_etc must return as its value the number of equations, neqns, the TRANSIENT's global variable nnodes must be set at the latest in the body of generate_grid, and Initialize and initNL3BANDV must pass through its arguments the pointers to certain arrays. And all the initializations that may depend on values changed in read_profiles must be deferred to auxiliary_parameters_for_evalf. Two other optional TRANSIENT's global variables, title and time_units, should also be defined preferably in get_neqns_etc so that they are available from the beginning in case that interactive help (?c-, ?ti, etc.) or the parameter lists (-d and -p command-line arguments) are required - these all are generated by read_input. Also the default (most often used) values of tStart, tEnd, tStepInit, tOutInit, tOutMulF, etc. and nOutOTH may be customized for the given problem by redefining them in get_neqns_etc, which may thus look like this:

int get_neqns_etc(void)
title = "                     TRANSIENT \n \
                Demonstration No. 4 \n \
time_units = "";
tEnd = 50.;
tOutInit = 0.1;
tStepInit = 1e-4;
nOutOTH = 1;
return NEQNS;

or like this:

int get_neqns_etc(void)
time_units = "sec";
tOutMulF= 1.6;
return NEQNS;

If equidistant grid is used, generate_grid can be very simple, in addition to the definition of nnodes, it suffice to define the grid step h (TRANSIENT does not need to know the grid positions xj; in pr_profiles, one can use the formula xj = x0 + j*h). A more complicated example of a nonequidistant grid is in demo1.c.

The value of nnodes may be changed in a suitable place in a PDM. However, it must never exceed the value initially set in generate_grid. Eqs. (1a) are then solved only on the subset of the initial grid from node 0 up the the current node nnodes - 1. This feature can be useful in modelling moving boundary problems, with a moving boundary at the RHS end of the model.

The function Initialize is primarily intended to initialize the concentrations. Memory for the "old", c_o[j*N+k]  = ck(j, l ), and "new", c[j*N+k]  = ck(j, l + 1), concentrations is allocated in TRANSIENT, and the pointers to c and c_o are passed into the PDM through the two arguments, cp and c_op, of Initialize. As discussed in Sec. 3, in a PDM these arrays can be declared as ROW *C, *C_o and one can refer to them using the matrix notation C[j][k], etc. The pointers must be properly typecast when passed between TRANSIENT and the PDM. Two statements of the following type have to appear in the body of Initialize:
        C = (ROW *)cp;
        C_o = (ROW *)c_op;
When neqns = 1, there is of course no need to define the ROW type and use the typecast operators (see demo2a.c, demo2b.c and demo3.c).

C_o[j][k] must be assigned the initial values corresponding to t = t0 for the given problem, and C[j][k] some suitable initial guess for the initial time step (C[j][k]   C_o[j][k] will usually do).

The function initNL3BANDV also plays the role of an interface, the interface between PDM and NL3BANDV. First, it transfers the addresses of the BDA arrays from the PDM to NL3BANDV as described in detail in Sec. 3.1. Second, it passes the addresses of the vector of the function values and of the Jacobian submatrices , , , (sharing memory with ), and (sharing memory with ) (see Sec. 2.4) from NL3BANDV to PDM. These matrices are declared in nl3bandv.c as pointers to double. If you wish to refer to them in PDM (mainly in evalf) as to matrices (ROW), define in the PDM a set of pointers to ROW, e.g., ROW *AROW, etc., and use again the type cast operator,
        A ROW = (ROW *)A double;
where Adouble is the formal argument of initNL3BANDVfor . Similarly for the and matrices. XROW may be introduced as an alternative pointer to AROW, similarly YROW = DROW. See PDMtemplate.c or individual demos for more details.

read_profiles can be used to overwrite any initializations in Initialize by reading the profiles from a file whose name is argument of this function. This is useful when an integration is to be continued with modified parameters, or after a system crash. read_profiles can simply be an "inverse" of the pr_profiles function described below. It can also be empty (take no action) if one plans to always start from the beginning.

The purpose of function auxiliary_parameters_for_evalf is to avoid repetitive calculations in evalf. Various subexpressions occurring repeatedly in evalf that need to be calculated only once can be declared as global variables and defined in auxiliary_parameters_for_evalf.

evalf calculates the Jacobian submatrices , , , , and , and the vector of the function values (see Sec. 2.4) for the j-th grid point (the only argument of evalf) at the current guess for concentrations, stored in the "new" C. Before a call to evalf, all elements of matrices to are zeroed. One thus has to set in evalf() only all the nonzero elements. This is not true for F.

Note that in demo1.c and demo4.c the three "mixed" solutions (j - 1), (j) and (j + 1) are stored in three vectors VM, V0 and VP (in some order). Pointers to these vectors are stored in cBarM, cBar0 and cBarP, such that cBarM points to the vector that contains (j - 1) for the current j, cBar0 points to (j) and cBarP to (j + 1). As j is increased by 1, these pointers are simply rotated (cBar0 cBarM cBarP cBar0) instead of moving the contents of the vectors, and only the vector pointed to by cBarP is updated.

The implementation of evalf in demo1.c (or in demo2a.c, demo2b.c and demo4.c) may hopefully be used without substantial changes as a template for many problems with simple diffusion terms of the type D ck/x2 to calculate the contribution of the diffusion terms and the time derivatives to , , , , , and . On the other hand, the contribution of the reaction terms Rk of eq. (1a) must generally be coded from scratch for each problem. In demo1.c it is calculated in two separate functions, reaction_part for all nodes except the last one, and reaction_part0 for the last node.

The function pr_profiles is called at time instants determined by tOutInit and tOutMulF, and its role is to output the current concentration profiles to the 'Output'* file(s), i.e., to the output stream out (using e.g. the PR macro defined in incl_pri.h). Its single argument contains the maximum number of iterations used in a single time step since the last invocation of pr_profiles. The pr_profiles functions in demo1.c to demo4.c also invoke report_max_resid defined in nl3bandX.c. If steady=1, pr_profiles is called only once to output the steady-state profiles with its argument set equal to the number of iterations needed for the last successful attempt.

The functions init_Other, update_Other and pr_Other can be used to calculate any supplementary time-dependent quantities depending on the concentrations (such as the corrosion current, its integral over time, and the corrosion potential in demo1.c; comparison of the numerical and analytical solutions in demo2a.c and demo2b.c; the surface roughness in demo3.c; or the total amount of species k present in the model at time t, i.e., the integral of ck over x), and write them out conveniently at the end of the par'Output' file (using e.g. the Pm macro defined in incl_pri.h). pr_Other is called nOutOTH times more often than pr_profiles. The argument of pr_Other is set equal to 0 when the profiles are simultaneously written onto 'Output', and equal to 1 when they are not. Its value may be used in the body of pr_Other to avoid duplication of some output, or on the other hand, to add output for an additional time instant - see more notes in PDMtemplate.c; both pr_Other and pr_profiles may actually send alternate output to any of the two streams, or anywhere else, if user wishes so.

If one needs to do additional output at an arbitrary time, instead of using pr_profiles() and pr_Other(OnlyOther), one should use the out_profiles(NewFile) and out_Other(OnlyOther) functions (see the beginning of transien.h; NewFile is either 0 or 1, directing the profiles either into the same file as the previous ones, or to the next "out" file). These two TRANSIENT functions first check whether the profiles or the "Other" values may have already been printed for the same time, thus preventing the duplication of output.

The function init_Other is invoked just before the transient or the steady-state calculation starts, and may be used for any initializations needed for the calculation of the supplementary quantities. For steady = 2, init_Other is thus called twice - see how demo1.c handles this. If profiles are initialized from a previous output file, init_Other may perform some initializations from the corresponding par'Output' file, see demo1.c for an example.

The function update_Other is called after the successful completion of each time step and may be used for example for integrating some quantities over time. update_Other should normally return 0. If pr_Other() is to be executed immediately after the termination of update_Other, e.g. when a certain condition requires additional output, this can be achieved by update_Other() returning a nonzero integer. Moreover, if it returns a negative number, i < 0, the execution is terminated immediately with the exit code 1000 - i. When the macro RESETtStepAT99 is set during the compilation (see the file defs.h), the i > 1 return values of update_Other() also have the following additional effect: if i = 99, the next time step is reset to its initial value of tStepInit, and for all other i > 1, it is decreased by the factor of 10i. At the same time, the ATSA is re-initialized.

One can also use update_Other() to reduce the magnitude of the next time step tStep explicitly to a certain value without re-initializing the ATSA. For example, if the value of a quantity BCv has to be changed at time t1, put in the body of update_Other() the following lines:
        static int t1Flag = 1;
        if(t1Flag) {
            if(t >= t1) { BCv = BCv1; t1Flag = 0; }
            else if(t+tStep >= t1) tStep = t1-t;
The else if condition gets satisfied first and as a result tStep is reduced so that the next time step ends exactly at time t1. During the next invocation of update_Other() (at the end of the next time step), the preceding if condition is satisfied, and the value of BCv changed to BCv1 exactly at t=t1. Here we assume that time is positive and increasing. t1 and BCv1 can be variable model parameters to appear in the PDM header file.

For a steady-state calculation, update_Other and pr_Other are called only once after the completion of the calculation.

All three functions init_Other, update_Other and pr_Other can be empty. If you plan to use TRANSIENT for both the steady-state calculation and the transient solution, you may wish to use the switch steady (exported from transien.c) in the functions pr_profiles, init_Other, update_Other and pr_Other to differentiate their actions for the two types of calculations (see the supplied demos). Always use the "old" solution c_o in all these output functions.

8. Demonstration examples

Four demonstration examples (demos for short) of TRANSIENT applications are provided. Each is located in a separate directory demo/demoX, X=1,2,3,4. Start exploring the demo directory tree by reading the demo/0readme.txt file. Demo 1 is a relatively simple example of the rather complicated systems that TRANSIENT can handle. However, it may be too complicated as a starting point for the exploration of this package. Demos 2 and 4 provide much simpler starting points.

Demos 1, 2 and to some extent (for some parameter values and/or time ranges) also demo 4, are types of problems when the evolution of concentration (dependent variable) profiles is uneven in time, and thus the use of TRANSIENT with its adaptive time-stepping algorithm is of big advantage for such problems. On the other hand demo 3 in its original formulation requires constant time step. Nevertheless, TRANSIENT seems to handle even such a problem quite satisfactory - it automatically keeps time step practically constant.

8.1. Demo 1. Uniform corrosion of a metal surface in contact with two consecutive porous media

The files demo1.h and demo1.c in demo/demo1 are an example of the PDM for the following problem with neqns = 6:

Here all the concentrations ci are functions of x and t. Initial conditions are


This set of equations describe diffusion-reaction, precipitation/dissolution and adsorption/desorption processes in two fully saturated porous media in contact at x = L1. is the porosity accessible to the solution, is the effective porosity for diffusion, and is the tortuosity factor. All these three quantities are constant in each medium, but their values are different in the two media. For 0 < x < L1, one has = , = , and = . For L1 < x < L1 + L2, one has = = and = . Unknowns c0, c1 and c3 are pore water concentrations of three different species, c2 and c4 are volume concentrations of precipitates, and c5 is a sorbant concentration. All ci must be continuous at the interface at x = L1. Also the diffusion fluxes across this interface must be continuous, which introduces a discontinuity in the first derivatives of c0, c1 and c3:

Naturally, ck(x, t) 0 for all k, and there is a piecewise constant upper bound for c5 given by eqs. (24).

The first medium is in contact with a uniformly corroding copper wall at x = 0, and this electrochemical corrosion process determines boundary conditions k, left for the diffusing species at x = 0. These involve the corrosion current related to the diffusion flux of the first two species:

where is the Faraday constant. This current further satisfies the following relation:

For c3, there is no diffusion flux at x = 0:

k, right at x = xright are constant concentration b.c.:


For the nondiffusing species, eqs. (21), (23) and (24) play also directly the role of the boundary conditions - these nondiffusing equations are not true partial differential equations because in them x only plays the role of a parameter.

The region near x = 0 is very important because the electrochemical reactions (eqs. (27) and (28)) that drive the whole process take place there. It is thus desirable to have a much finer grid near x = 0 than near xright. To this end a "geometrical progression" grid is used, with xj+1 - xj = f (xj - xj-1) where f is constant for all j, i.e.,

Then = f  in eqs. (4) and (5), and this f is also identical with the f of eqs. (6).

The input for generate_grid in demo1.c are the numbers of grid intervals in (0, L1), interv1, and in (L1, L1 + L2), interv2. generate_grid then determines x1 and f such that one grid point lies exactly on the interface: xinterv1 L1. Note that for the geometrical progression grid, all coefficients on the right-hand side of eq. (4) are simply multiplied by 1/f2 (f2 is called fsq in demo1.c) when j is increased by 1. These coefficients are stored in demo1.c in variables ja2 and jd2.

For the internal nodes in the two media, we can simply divide eqs. (19), (20) and (22) by or and get a set of equations of type (1a) and discretize them as described in Sec. 2.2 or Sec. 2.3 - for the internal nodes both discretization approaches will give identical results. At j = interv1, the "coefficients" (, etc.) in eqs. (19-24) have a discontinuity, and to derive the discretized Fkinterv1, the best way is to use the mass-balance approach of Sec. 2.3, i.e., to start from the integral form of the mass-balance equations (19-24) for the small region (L1 - /2, L1 + /2). Note that after dividing the diffusing equations (19), (20) and (22) by , their reaction terms can be written in the form

Rk = + pk ,

where rk and pk are not explicitly dependent on x, and is either or . Similarly, for the remaining nondiffusing equations, the form is

Rk = rk +  pk .

We can assume that ck(x, t) (and thus also rk and pk) are constant in the tiny interval (L1 - /2, L1 + /2) and equal to their values at L1, ck(x, t) = ck(xj, t) = ck(j, l + 1); j = interv1. Integrating then eqs. (19), (20) or (22) from L1 - /2 to L1 + /2) gives

Here j = interv1, and the continuity condition of eq. (26) for the diffusion flux density was used at x = L1. Using now in eq. (30) discretization formulas (2), (7a) and (7b) gives

Here j = interv1, and

In demo1.c, eps[2].

Note that the reaction term in eq. (31) has exactly the same form as at the interior nodes, except that is replaced by . The same result is obtained for the nondiffusing eqs. (21) and (23). Discretizing these equations in the same way as the diffusing ones above, one gets

For the j = 0 node (LHS boundary conditions), the 3-point formulas (6a) were used directly in eqs. (27-29) without much thought when TRANSIENT was first used to study systems of this type in 1995. The resulting discretized equations then served as the expressions for Fk(0). This was implemented in the old demo 1 source file. The old approach did not enforce mass-conservation within the surface region (0, x1/2). However, this region is usually very tiny, and thus the resulting error could be expected to be rather small.

In the meantime I have switched over to using the more consistent mass-balance discretization also in the surface region even in cases much more complicated than that eq. (28). Such an approach is finally implemented also in the current demo1.c. Mass-balance discretization of the "boundary conditions" should give better results whenever the diffusion fluxes (slopes of ck) are high in the surface regions. In the demo 1 case, the only noticeable (but quite small) difference between the old results and the current ones is in the corrosion potential in the first 150 years (see below).

In the mass-balance approach for the j = 0 node, one has to integrate eqs. (19-24) also in (0, x1/2). The result of this integration can be obtained from eq. (30) after replacing in it by or as appropriate, by , xj by 0, by 0, and by x1. For (, t), one can again use eq. (7a), but for (0, t), eqs. (27) and (29) must be used. This gives

Here k = 0, 1, 3; is the diffusive flux density for the k-th species at x = 0. From eq. (27) one has

Finally, icorr has to be expressed as a function of c0(0, t) and c1(0, t) by solving eq. (28). It is a cubic equation in icorr, and one can show that it always has a single nonnegative real root (because A1 0, A2 0), which gives the nonnegative icorr. It is however easier to solve it numerically using Newton's method, than to use the analytical formulas expressing the real roots of a cubic equations in terms of complex numbers. Such numerical solution is implemented in demo1.c in the function iCorr_f (to be used in init_Other() and pr_Other()), and also at the beginning of the j = 0 branch in evalf(), where in addition to icorr also the derivatives of icorr with respect to c0 and c1 are needed.

In demo 1, init_Other calculates the initial value of icorr, update_Other calculates the time integral icorrdt, and pr_Other outputs to the par'Output' file both of these quantities and also the corrosion potential

A possible way to reduce the number of operations in evalf is to include the factor in the definitions of all Fk(j). This will eliminate the factor New=1 - (see the paragraph below eq. (12)) from all terms of all non-zero elements of the Jacobian matrices A, B, D, X and Y, and add the factor 1/New only to the contributions of the time derivatives to the diagonal elements of B. This is not implemented in demo1.c, but it is recommended for more complicated systems with a large number of non-zero elements in the Jacobian matrices.

Names of variables used in demo1.c closely follow the notation used in eqs. (19-25) (e.g., km11 represents k-11, etc.).

Demo 1 sample results obtained with TRANSIENT V3.0 are in the directory demo/demo1/out. However, they do not differ substantially from the older V2.51 results, retained in the demo/demo1.V2.51/out. Thus the rest of this section was left unchanged, referring to the old V2.51 results:

Discussion of the sample results from demo/demo1.V2.51/out (August 2001)

An example of input data for the steady-state calculation is the file dataS1. The corresponding output is in files S, parS, etc. Number of iterations needed to obtain this steady state may be different on different machines because even a small difference in the precision of the floating-point arithmetics may show up. If more than one attempt was made, a much larger difference in the number of iterations may be observed due to possibly different implementations of the basic random number generator rand, which is used to generate initial guess for individual attempts at calculating the steady state. Of course, the resulting steady state must be (within the specified tolerance) identical on different machines.

The steady state obtained with dataS1 is stable. One can check this by using this state as the initial state for a transient calculation (e.g., change steady 1 in dataS1 to steady 2, and rerun for an arbitrary tEnd). However, for the parameter values used, it takes about 390 million years (of real time t) to reach this steady state from the initial state of eq. (25).

An example of the input data file for transient calculation is data1. Model parameters are the same as in the steady-state input file dataS1. To get some interesting results, one has to specify tEnd larger than 4,733,640,000 sec (150 a). This is the time when the oxygen concentration (c0) drops to zero at x = 0. Therefore corrosion stops (icorr = 0 by eq. (28)) at that moment. Within about the next 50 a, all oxygen that was initially present in the first, rather porous, medium (clay) is used up. The second medium represents a high quality rock with a very small porosity through which additional oxygen can diffuse extremely slowly from groundwater flowing relatively fast through a major fracture in the rock assumed to be situated at the RHS boundary (i.e., the boundary conditions at x = xright = L1 + L2 simulate the surface of a large fracture). The dissolved Cu(II) concentration (c3) reaches a maximum at x = 0 in about 30 a. It then slowly decreases and spreads from the corroding surface. Up to the time of about 5000 a, buildup of the Cu(I) concentrations (dissolved c1 and precipitated c2) takes place. At about 350,000 years, oxygen concentration drops to zero throughout the whole model except for the tiny vicinity of the fracture zone at the RHS boundary. After that, it takes hundreds of millions of years before the excess c1 diffuses out of the system through its RHS boundary before the steady state is being approached. During this time, all oxygen coming in through the x = xright boundary is consumed in the nearest vicinity of xright, and c0 = 0 in the rest of the system. Thus also icorr = 0.Only when the steady state is finally being approached, nonzero c0 front again reaches x = 0, and very slow corrosion represented by a tiny icorr resumes. Comparing parLONG and LONG33 with parS and S, one can see that the corrosion potential settles to the steady-state value of -0.731 V at about 390 million years. At the same time the concentration profiles also assume the form of the steady-state from file S.

Increasing interv1 and interv2 (decreasing internode spacings) above the values in data1 can still improve some details of the profiles for some ck.

Validation of the demo 1 results by comparison with a crude analytical model

Certain observations about the nature of the numerical solution at long times are used here to construct an analytical model confirming the validity of the demo 1 results (time needed to reach the steady state).

Note that in the time interval from about 350,000 a to about 180×106 a (output files LONG22 to LONG31), the controlling process is the slow diffusion of Cu(I) out of the model. One can verify this by solving only eqs. (20) and (21) with c1 and c2 initialized with the results of the LONG run for t = 349.5×103 a (i.e., from the file LONG22). To achieve this, run demo1 with the input file data1cu. Sample output from such a run are the files parLONGcu, LONGcu1 and LONGcu7.

demo1.c is now capable, in addition to solving the full model with six species, also to solve its two simpler subsets. The new input parameter onlyCuI serves as the switch between these three cases: onlyCuI=0 gives the full model; onlyCuI=1 solves eqs. (20) and (21) only (with all other concentrations set to 0); and onlyCuI=2 solves eq. (20) only, with k5 = 0, after all the initial c2 was converted instantaneously into c1 (c1 + c2/ c1). The values onlyCuI=1 and 2 are permitted only with the initialization from a previous profiles file. In data1cu, onlyCuI=1 and init_from_file=LONG22.

Comparing the file LNGcui (the two species subset solution) with LONGi+22, i=1,2,...,8, shows that they both give exactly the same c1 and c2 values (3rd and 4th columns). c1 starts to differ near x = xright only between files LNGcu9 and LONG31 (i.e., for t = 179×106 a, when nonzero c0 starts to spread from x = xright into the interior).

This excellent agreement is due to the fact that the two species subset correctly represents the dynamical equilibrium between the dissolved and precipitated Cu(I). Somewhat worse agreement is obtained with onlyCuI=2 when all c2 is instantaneously converted into c1 at the beginning of the run.

Note further that between about 2.7×106 a and 180×106 a, c1 is practically constant in the first medium, and in the second medium it decreases almost linearly to zero. Thus in this time interval, one can approximate c1 roughly as follows:

This is true for the full model, as well as for its two simplified subsets. Similarly, c2 is constant in the first medium, and very close to 0 in the second medium:

Using these approximations, the total mass of the two Cu(I) species contained in the model is

If precipitation/dissolution is faster than diffusion (D1), c2(0, t) is approximately related to c1(0, t) through the steady state solution of eq. (21):

Indeed, this relation is well satisfied by the results of the LONG run.

M(t) can change only as a result of the diffusion of c1 through the RHS boundary. This diffusion outflow is proportional to the slope of c1(x, t) (at xright):

Using eqs. (34) and (35), the solution of eq. (36) is


and tsat is the time, when c1(0, t) drops to the c1sat value, i.e.,

and for t tsat,

The following table compares the numerical values of c1(0, t) obtained in the LONG run (using input data file data1) with the approximate analytical results of eq. (37).

  c1(0, t) (mol/m3)
Time (106 a) Full model Eq. (37)
2.796203 (tlin0) LONG25 0.245 0.245
5.592405 LONG26 0.214 0.222
11.18481 LONG27 0.167 0.182
22.36962 LONG28 0.102 0.122
44.73924 LONG29 0.0343 0.0476
89.47849 LONG30 0.00392 0.00648
178.9570 LONG31 5.45×10-5 1.20×10-4
357.9139 LONG32 1.37×10-8 4.13×10-8
382.7401     1.37×10-8
The beginning of the time interval in which c1 is nearly linear in the second medium, tlin0, was set to the time corresponding to the LONG25 file, tlin0 = 2.796203×106 a, and tsat = 28.07653×106 a.

Given the nature of the approximations used to obtain eq. (37), the agreement of the two solutions in the above table is rather good, confirming that the TRANSIENT based programs give realistic time scales for the modeled processes. The fact that eq. (37) gives solution that decreases slower with time than the numerical solution, only further confirms a good performance of TRANSIENT. Namely, eq. (33a) does not strictly satisfy eq. (20). Compared to eq. (33a), the true c1 is a slightly convex function, the slope of which at x = xright is more negative than that of eq. (33a). Steeper slope results in faster diffusion of c1 out of the system, and thus somewhat shorter duration of the modeled processes.

8.2. Demo 2. Gas generation and diffusion

This is a rather trivial application of TRANSIENT. It is a case for which some analytical results are available, and can thus be compared with the numerical values obtained by TRANSIENT. Different handling of an interface in the interior of the system is discussed here (see the input parameter INTERFACE). This is also related to the type of the grid used. The two demonstration PDM's in the subdirectory demo/demo2, demo2a.c and demo2b.c differ only in the spatial grid used. Because neqns = 1, this case allows to compare the speed of nl3band1.c with that of nl3bandv.c - to link demo2a.c or demo2b.c with nl3bandv.c instead of nl3band1.c, no changes need to be done in the PDM.

The set of equations solved in this example is:

It corresponds to the following idealized physical system: hydrogen is generated by radiolysis at a constant rate G inside a cylindrical container that is originally (at t = 0) filled only with humid air. The length of the container is l2 and its cross-section A2. The container is closed on the right and has a (narrow) concentric inlet of length l1 and cross-section A1 on the left. Let us put the coordinate origin at the open end of the inlet. The inlet extends from x = 0 to x = l1 and the container from x = l1 to x = l1 + l2. Let c(x, t) be the concentration of hydrogen. It is assumed that this concentration does not depend on the distance from the long axis of the system, and so one can use the one-dimensional equations (38-41). Hydrogen that diffuses through the open end of the system at x = 0 is dispersed instantly by air flow in the surrounding atmosphere (cf. the first condition in eq. (39)). D is the diffusion coefficient of hydrogen in the air, and eq. (40) expresses the continuity of diffusion flux at the interface (the effects of flow constriction are neglected). If A1 A2, the slope of c is discontinuous at l1.

Possible questions to be answered: Does the maximum steady-state concentration exceed the critical explosive concentration of hydrogen in the air? If so, how long does it take to reach the critical concentration? To answer the second question we need to know the transient solution.

Analytical steady-state solution of eqs. (38-40) is

demo2a.c and demo2b.c compare directly this analytical solution with the numerical one. They output the analytical steady-state solution (42) as an additional column in the steady-state output file.

The transient solution is more complicated. It is easy to get a closed formula for the Laplace transform,

of c(x, t) satisfying eqs. (38-41):


To invert this Laplace transform analytically does not seem to be possible. Thus when steady = 0, demo2a.c and demo2b.c calculate k(x, s) numerically (using eq. (43) in update_Other) for a single value s in each run (s is an input parameter). Using then pr_Other, this numerical k(x, s) is outputted as a function of x together with the analytical solution (44) at the end of the parameter output file. In this way, one can check the accuracy of TRANSIENT at least in terms of k(x, s). Remember that large values of s probe c(x, t) for very short times only; very small values of s must be used to probe the long time behavior.

Note that eq. (43) involves integration to t = . TRANSIENT gets numerical results for c(x, t) only for t (0,). pr_Other assumes when calculating the tail of integral in eq. (43) that a steady state was already achieved at t = , i.e., that c(x, t) = c(x,) for t > . If it seems that the steady state has not yet been reached (that the last two profiles are not identical), a warning appears in the par'Output' file. For the purpose of the above comparison, make sure that tEnd is large enough. Remember that for small tEnd and small s, the "numerical" column for k(x, s) in par'Output' is wrong due to the above tail approximation.

Discretization of the interface condition (40):

They are two possibilities:
(i) The standard (simple) approach is to approximate the left and right derivatives in eq. (40) by


where j is the index of the grid node nearest to the interface at x = l1, and use the result directly for F0(j).
(ii) The alternative approach is to use the mass-balance discretization of Sec. 2.3 in the vicinity of the interface, i.e., in the interval (xj - /2, xj + /2) using condition (40) at the interface - cf. eqs. (30) and (31) of demo1 and the accompanying discussion.

Approach (i) was not used in demo1 at all. Here it is included because the existence of the (semi)analytical solution makes it possible to compare it quantitatively with approach (ii). The switch between the two approaches in demo2a.c and demo2b.c is provided by the input parameter INTERFACE. Comparing the results obtained with different values of INTERFACE, one can see that approach (ii) always gives better results, as expected. The RHS of eqs. (45) are indeed poor approximations for the left and right derivatives at the interface, but they are natural approximations for the derivatives at midpoints between the nodes (cf. eq. (7a) and (7b)).

demo2a.c uses a single equidistant grid for the whole region 0 x l1 + l2. This is obviously not the best choice, as in such a grid the interface at x = l1 will not generally coincide with a grid point. It is used to explore the consequences of such a bad choice. Let h be the distance between the two neighboring nodes, j the index of the node nearest to the interface at l1, and = l1 - xj (|| h/2) the difference in the positions of the interface and the nearest grid node.

In this case = = h in eqs. (45). When INTERFACE = 2 or 3, condition (40) is used directly (approach (i)) and F0(j) of eq. (8) is

When INTERFACE = 0 or 1, eqs. (38) are integrated from xj - h/2 to xj + h/2 (approach (ii)), and

Here A1 and A2 have the same role as and in demo 1 - these are all different kinds of "capacity factors" equal to the volume available for mass transport (storage of diffusants) per unit length of the one-dimensionalized systems.

Apparently, the larger the ||, the poorer the agreement one can expect between the numerical and analytical results, because the change in the slope of c occurs at the interface, not at the node xj. To verify this, INTERFACE = 0 and 2 correspond to the original system as described above, whereas INTERFACE = 1 and 3 to a modified system in which the lengths l1 and l2 are adjusted (changed to l1' xj = l1 - and l2' = l2 + ) so that the total length of the system remains the same, and a grid point lies on the interface and thus = 0.

The supplied sample input file demo/demo2/data2 corresponds to INTERFACE = 0. In this case 0, and the agreement between analytical solution and TRANSIENT's numerical results is not good enough both for the steady-state and transient solutions. If you switch to INTERFACE = 1 (adjust lengths) in the input file, you will get a perfect agreement for the steady state (in all decimal digits printed) and the best agreement possible (better than for all the other values of INTERFACE) for the transient solution.

If you switch to INTERFACE = 2 or 3 (eq. (46); approach (i)), the agreement is much worse than when using eq. (47), both for the modified and the original systems.

The conclusion is: for the best results, use the mass-balance discretization at the interface, and put a grid point on the interface.

Adjusting length as in demo2a.c (INTERFACE = 1 or 3) may not always be an ideal solution when the exact dimensions of the system do matter. It results in the best numerical solution, but for a slightly different system than the specified one. That is why in demo2b.c a nonequidistant grid is tested that always has a grid point at the interface for arbitrary l1 and l2. Everything else is the same as in demo2a.c.

demo2b.c uses only two different grid steps: h1 in (0, l1) and h2 in (l1, l1 + l2). Then = h1 and = h2, and eq. (46) is replaced by

and eq. (47) by

Here j is the index of the interface node.

For compatibility with demo2a.c, INTERFACE can again be set to all four values 0 to 3, but 0 and 1 (and 2 and 3) will give identical results (now there is nothing to adjust when INTERFACE = 1 or 3). Again, INTERFACE = 0 (mass-balance discretization used at the interface; approach (ii)) gives perfect or near-perfect agreement between numerical and analytical concentrations, whereas INTERFACE = 2 (direct continuity-of-flux condition; approach (i)) gives much poorer agreement.

Alternatively, one could also use the "geometrical progression" grid as in demo1.c, especially when l1 l2 (to have enough points in the important inlet region where concentration changes rapidly, without introducing a very large difference in the neighboring grid steps at the interface).

Finally, in both demo2a and demo2b there is a possibility for the last node to switch between the 3-point discretization of eq. (6b) for the second condition of eq. (39), and the mass-balance discretization, given for demo2a by

For demo2b, h is replaced by h2. The input parameter BCr (right b.c.) having legal values `3point' and `massBal', switches between these two options. Because c is practically constant in a rather large vicinity of x = l1 + l2, both approaches give in this case identical results (for a rather different situation, see demo4).

8.3. Demo 3. Stochastic surface growth

The following single equation for the dependent variable h(x, t) is solved:

This equation is used in ref. [15] as a very simple model for growing fractal surfaces. Here h(x, t) represents the height of a one-dimensional surface at point x and time t. It is the last, singular term of eq. (50), that for certain values of parameters A and B, is responsible for the fractal character of the growing surface.

What is actually studied in ref. [15] is the discretized version of eq. (50) corresponding to = 1 (fully implicit scheme), the unit grid step, i.e., xj jj = 0, 1,..., L, and periodic boundary conditions. TRANSIENT cannot at present handle periodic boundary conditions (this would require a modification of nl3bandX.c that would almost double the size of its work memory). Instead, in demo3.c there is a choice of von Neumann (input parameter bc = 0) or Dirichlet (bc = 1) boundary conditions identical at the two ends of the system. In both cases, the boundary value is supplied through the input parameter bcValue. Because of this difference in the boundary conditions, one cannot expect that the character of the surfaces generated by TRANSIENT will be exactly the same as in ref. [15].

Also in ref. [15] the discretized finite difference equation does not correspond exactly to eq. (50) (eq. (4) of ref. [15]). In ref. [15], the factor was omitted in the difference formula for . This has the same effect as multiplying A by 2. In demo3.c, eq. (50) is discretized consistently (using the unit grid step again). Thus, A = 0.001 of demo3.c corresponds to A = 0.002 in the figure captions of ref. [15], etc.

The functions init_Other and pr_Other are used to calculate the "roughness" of the surface defined as

Here < h > = h(j, t), etc. The function pr_profiles does not output plain h(j, t), but rather  (h(j, t) - < h > ) so that one can compress a lot of surfaces for different times into a single plot, and enhance their roughness at the same time. SF and CF are input parameters of demo3.c.

With = 1 and bc = 0 (von Neumann), TRANSIENT can generate surfaces resembling closely those of ref. [15] (see data3 in subdirectory demo/demo3).

As is decreased, the generated surfaces can change considerably. Also any slight change in the adaptive time-stepping algorithm during TRANSIENT's evolution usually led to a big change in the appearance of the generated surfaces. This is illustrated by the enclosed figures (files and which compare the results obtained for w(t) with four different previous versions of TRANSIENT. It confirms the claim of ref. [15] that the features of the surfaces produced are the consequence of the discretized finite difference equation, not of the original eq. (50).

The Dirichlet boundary conditions (bc = 1) will anchor the surface at both ends, and thus generate a very different type of surface: a smooth growing "tip" (see data3.tip input file) that ends up in a steady state.

Steady-state calculation for bc = 0 will always fail (even if one changes the code in demo3.c to initialize h(x) directly to the trivial steady state h(x) = (1 - A)x and specifies bcValue  =  1 - A). This can be proved exactly: in this special case the matrix of eq. (15) is always singular when bc = 0, irrespective of the initial guess for h(x). The same is true when the 3-point difference formulas used in the boundary conditions are replaced by the 2-point formulas, or when different boundary values (slopes of h(x)) are used at the two ends.

8.4. Demo 4. Brusselator

The nonlinear Brusselator model [16] is often used in mathematical biology to study morphogenesis (e.g., [17]). It consists in the following two equations:

Here X and Y are the concentrations of two morphogens - an activator (X) and an inhibitor (Y). In demo4.c, eqs. (51) are solved in the interval (0, L) with the zero-flux von Neumann boundary conditions = = 0 at both ends of (0, L).

Rate constants a, b, c and d of eqs. (51) correspond in demo4.c to input parameters aa, bb, cc and dd.

The zero-flux b.c. at both model ends are again handled in two ways: directly using the 3-point discretization formulas of Sec. 2.2 or the mass-balance discretization of Sec. 2.3. The input parameter BC having legal values `3point' and `massBal' is provided to switch between these two options. The concentration profiles obtained with BC = 3point and BC = massBal differ significantly near x = 0 and x = L when X and Y vary rapidly with x in these regions, which they often do. Thus in this case the improvement introduced by the use of the full mass-balance equation at the boundary nodes may be significant. On the other hand, when the concentration profiles are nearly constant near the end of the domain of eq. (1a) (such as in demo 2), both approaches give the same results.

The behavior of the solutions of eqs. (51) can vary considerably depending on the values of the parameters involved. One can get an idea of what type of transient behavior to expect for a given set of DX, DY, a, b, c and d by performing the stability analysis of the linearized eqs. (51) (these linearized equations are called the Turing model, first studied by A. M. Turing [18] of the Turing machine fame).

There may be more than one steady state (there is always at least the trivial steady state X(x) = a/d, Y(x) = bd /ac). For example with TRANSIENT V1.3, the default parameter values (when starting demo4 by 'demo4 outfile_name', then typing 'steady 1', then 'end') gave a steady state consisting of two periods of a sine-like wave. The same parameters gave transient solution in which the initial random nonuniformity in space is slowly amplified until a single peak in X (corresponding to a minimum in Y) develops. With TRANSIENT V2.04, one got the trivial steady state in both cases.

On the other hand, when data4 was used as the input file in TRANSIENT V2.04, the results went fast to a single-peak steady state, whereas with TRANSIENT V1.3, the initial nonuniformity disappeared rather quickly, and then the spatially uniform X and Y seemed to slowly approach the trivial-steady-state values. For the same input parameters, the three steady-state input files data4.s1, data4.s2 and data4.s3 differ only in the initial random guess for X and Y. They usually give three different steady states - the first one is the trivial steady state, the two other are mirror images of each other (or S2 can be identical with S3, depending on the machine); see the files in demo/demo4/out.


[1] R. Spotnitz, "NL3BAND", Technical Software Distributors, Charlotte, NC, 1994.

[2] J. Newman, "Electrochemical Systems", Prentice-Hall, Englewood Cliffs, N, 1973.

[3] R.E. White, "On Newman's Numerical Technique for Solving Boundary Value Problems", Ind. Eng. Chem. Fundam. 17, No. 4, 1978.

[4] T.V. Nguyen and R.E. White, "A Finite Difference Procedure for Solving Coupled, Nonlinear, Elliptic Partial Differential Equations", Comput. Chem. Engng. 11, 543-546, 1987.

[5] M. Matlosz and J. Newman, "Solving 1-D Boundary-Value Problems with BandAid: A Functional Programming Style and a Complementary Software Tool", Comput. Chem. Engng. 11, 45-61, 1987.

[6] A.K. Runchal, "Theory and application of the PORFLOW model for analysis of coupled fluid flow, heat and radionuclide transport in porous media", In: Coupled Processes Associated with Nuclear Waste Repositories, Ed. C-F. Tsang, Academic Press, pp. 495-516, 1987 (see also

[7] F. King and M. Kolá, "Prediction of the Lifetimes of Copper Nuclear Waste Containers under Restrictive Mass-Transport and Evolving Redox Conditions", CORROSION/95, paper No. 425, NACE International, Houston, TX, 1995.

[8] F. King, M. Kolá and D.W. Shoesmith, "Modelling the Effects of Porous and Semi-Permeable Layers on Corrosion Processes", CORROSION/96, paper No. 380, NACE International, Denver, Colorado, 1996.

[9] M. Kolá and F. King, "Modelling the consumption of oxygen by container corrosion and reaction with Fe(II)", In: Proc. of the XIXth Int. Symposium on Scientific Basis for Nuclear Waste Management, Boston, Nov. 1995 (Mat. Res. Soc. Sym. Proc. 412, 547, 1996).

[10] F. King and M. Kolá, "A numerical model for the corrosion of copper nuclear waste containers", In: Proc. of the XIXth Int. Symp. on Scientific Basis for Nuclear Waste Management, Boston, Nov. 1995 (Mat. Res. Soc. Sym. Proc. 412, 555, 1996).

[11] F. King, M. Kolá, S. Stroes-Gascoyne, P. Bellingham, J. Chu and P.V. Dawe, "Modelling the Activity of Sulphate-Reducing Bacteria and the Effects on Container Corrosion in an Underground Nuclear Waste Disposal Vault", In: Proc. of the XXIIth Int. Symp. on the Scientific Basis for Nuclear Waste Management, Fall 1998 (Mat. Res. Soc. Sym. Proc. 556, 1167, 1999).

[12] F. King, M. Kolá, and D.W. Shoesmith, "Modelling the oxidative dissolution of UO2", In: Proc. of the XXIIth Int. Symp. on the Scientific Basis for Nuclear Waste Management, Fall 1998 (Mat. Res. Soc. Sym. Proc. 556, 463, 1999).

[13] F. King and M. Kolá, "Mathematical implementation of the mixed-potential model of fuel dissolution. Model Version MPM-V1.0", Ontario Hydro Report, 06819-REP-01200-10005-R00, March 1999.

[14] F. King and M. Kolá, "The copper container corrosion model used in AECL's second case study", Ontario Power Generation Report, 06819-REP-01200-10041-R00, Sept. 2000.

[15] M. Viscsek and T. Viscsek, "Intermittency, Stochastic Growth and Phase Transition in a Simple Deterministic Partial Differential Equation with a Singular Term", J. Phys. A: Math. Gen. 28, L311-L316, 1995.

[16] G. Nicolis and I. Prigogine, "Self-Organization in Nonequilibrium Systems", Wiley, New York, 1977.

[17] L.C. Harrison and M. Kolá, "Coupling Between Reaction-Diffusion Prepattern and Expressed Morphogenesis. ...", J. Theor. Biol. 130, 493-515, 1988.

[18] A.M. Turing, Phil. Trans. R. Soc. Lond. B237, 37, 1952.

Appendix: TRANSIENT 4.3-beta "packing list"

A complete list (map) of all the files in all subdirectories of the TRANSIENT 4.3-beta distribution package is in the HTML file index.html which can be opened in a Web browser, and from which one can directly inspect the contents of all the files.

TRANSIENT "engine":
transien.c The main, control, module.
defs.h TRANSIENT configuration file.
Header file to be included into a problem definition module (PDM): declares global variables exported from TRANSIENT, prototypes of the functions to be defined in the PDM, and functions to handle PDM-dependent input parameters.
CTRLpar.h List of control input parameters and their description.
I/O functions:
InOut.c File handling, input processing, output of control input parameters.
transien.h I/O for PDM-dependent input parameters.
Auxiliary modules:
DateTime.c A function returning the current date and time (to time-stamp the output files).
DateLen.h A header file for InOut.c and DateTime.c .
incl_pri.h Common #includes and printf macros.
Handles interrupt catching to be able to complete properly all output files in case of abnormal termination.
NL3BANDV package:
nl3bandv.c NL3BANDV solver for system of equation (8a).
matrixv.c Operations on linearly stored matrices.
lufacv.c LU factorization of linearly stored matrices.
matinvv.c Matrix inversion of linearly stored matrices.
A fast variant of nl3bandv.c for neqns = 1. Depends only on dynarr.c (and dynarr.h and defs.h).
dynarr.c Dynamical array allocation.
nl3_ApBo.h | These seven include files contain common code blocks for
nl3_conv.h |     nl3bandv.c and nl3band1.c .
nl3_Decl.h | In particular, nl3_conv.h contains
nl3_Rand.h |     the NR convergence criteria that
nl3_runb.h |     can be customized by the user,
nl3_strt.h |     if needed.
nl3_Decl1.h |

Headers for NL3BANDV package:
matinvv.h Used in nl3bandv.c .
lufacv.h Used in matinvv.c .
Used in nl3bandv.c, nl3band1.c, matinvv.c, lufacv.c, transien.h, and transien.c .
makefile UNIX makefile.
makefile.VisualCpp MC S Visual C++ makefile.
OBJECTS Microsoft C make file.
MAKEDEMOS.COM DCL command procedure for VMS.
index.html Web-browser entry point (map of the whole package in the HTML format).
00readme.txt Main plain text readme file.
doc/PDMtemplate.c Template for PDM source files.
doc/controls.txt List, description, and default values of control input parameters.
doc/function_lists.txt List of all functions declared in all TRANSIENT's modules.
doc/history.txt TRANSIENT's history.
doc/0license.txt License agreement.
doc/nl3_conv.h.V4.1 For comparion: nl3_conv.h for version V4.1
doc/nl3_conv.h.V4.2 And for the preceeding version V4.2
[doc/transien.tex] This manual. LATEX2e and LATEX2HTML ready. Uses some color.
Sorry, transien.tex is currently missing from the V4.3-beta distribution archive.

demo subdirectory: Demonstration PDM:

Contains a 0readme.txt file and four subdirectories each hosting one demonstration application of TRANSIENT: header and source files, makefiles, sample input files and a directory out with sample output and another readme file. You can conveniently browse through all this, starting from the above index.html using a web browser.

On Linux/UNIX, the case of the file names must be preserved (restored) as shown above. These are file names used in the #include directives and the makefiles.

(When it is not done explicitly, throughout this Manual any reference to Unix or Linux operating system should be understood as referring to both of them.)

Miroslav Kolar
2005-10-21, minor update 2023-11-04