NAG Library Routine Document

d03pdf  (dim1_parab_coll_old)
d03pda (dim1_parab_coll)

1

Purpose

2

Specification

2.1

Specification for d03pdf

2.2

Specification for d03pda

3

Description

4

References

5

Arguments

1:     $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system to be solved.

Constraint: $\mathbf{npde}\ge 1$.

2:     $\mathbf{m}$ – IntegerInput

On entry: the coordinate system used:

$\mathbf{m}=0$

Indicates Cartesian coordinates.

$\mathbf{m}=1$

Indicates cylindrical polar coordinates.

$\mathbf{m}=2$

Indicates spherical polar coordinates.

Constraint: $\mathbf{m}=0$, $1$ or $2$.

3:     $\mathbf{ts}$ – Real (Kind=nag_wp)Input/Output

On entry: the initial value of the independent variable $t$.

On exit: the value of $t$ corresponding to the solution values in u. Normally $\mathbf{ts}=\mathbf{tout}$.

Constraint: $\mathbf{ts}<\mathbf{tout}$.

4:     $\mathbf{tout}$ – Real (Kind=nag_wp)Input

On entry: the final value of $t$ to which the integration is to be carried out.

5:     $\mathbf{pdedef}$ – Subroutine, supplied by the user.External Procedure

pdedef must compute the values of the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$ and $U_x$ and must be evaluated at a set of points.

The specification of pdedef for d03pdf is:

Fortran Interface

Subroutine pdedef ( npde, t, x, nptl, u, ux, p, q, r, ires) Integer, Intent (In) :: npde, nptl Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, x(nptl), u(npde,nptl), ux(npde,nptl) Real (Kind=nag_wp), Intent (Out) :: p(npde,npde,nptl), q(npde,nptl), r(npde,nptl)

C Header Interface

#include <nagmk26.h> void  pdedef (const Integer *npde, const double *t, const double x[], const Integer *nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires)

The specification of pdedef for d03pda is:

Fortran Interface

Subroutine pdedef ( npde, t, x, nptl, u, ux, p, q, r, ires, iuser, ruser) Integer, Intent (In) :: npde, nptl Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, x(nptl), u(npde,nptl), ux(npde,nptl) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: p(npde,npde,nptl), q(npde,nptl), r(npde,nptl)

C Header Interface

#include <nagmk26.h> void  pdedef (const Integer *npde, const double *t, const double x[], const Integer *nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires, Integer iuser[], double ruser[])

1:     $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

2:     $\mathbf{t}$ – Real (Kind=nag_wp)Input

On entry: the current value of the independent variable $t$.

3:     $\mathbf{x}\left(\mathbf{nptl}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: contains a set of mesh points at which $P_{i,j}$, $Q_i$ and $R_i$ are to be evaluated. $\mathbf{x}\left(1\right)$ and $\mathbf{x}\left(\mathbf{nptl}\right)$ contain successive user-supplied break-points and the elements of the array will satisfy $\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{nptl}\right)$.

4:     $\mathbf{nptl}$ – IntegerInput

On entry: the number of points at which evaluations are required (the value of $\mathbf{npoly}+1$).

5:     $\mathbf{u}\left(\mathbf{npde},\mathbf{nptl}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$ where $x=\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nptl}$.

6:     $\mathbf{ux}\left(\mathbf{npde},\mathbf{nptl}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{ux}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ where $x=\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nptl}$.

7:     $\mathbf{p}\left(\mathbf{npde},\mathbf{npde},\mathbf{nptl}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{p}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must be set to the value of $P_{\mathit{i},\mathit{j}}\left(x,t,U,U_x\right)$ where $x=\mathbf{x}\left(\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$, $\mathit{j}=1,2,\dots ,\mathbf{npde}$ and $\mathit{k}=1,2,\dots ,\mathbf{nptl}$.

8:     $\mathbf{q}\left(\mathbf{npde},\mathbf{nptl}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{q}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $Q_{\mathit{i}}\left(x,t,U,U_x\right)$ where $x=\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nptl}$.

9:     $\mathbf{r}\left(\mathbf{npde},\mathbf{nptl}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{r}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $R_{\mathit{i}}\left(x,t,U,U_x\right)$ where $x=\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nptl}$.

10:   $\mathbf{ires}$ – IntegerInput/Output

On entry: set to $-1\text{ or }1$.

On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, d03pdf/d03pda returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.

11:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

12:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

pdedef is called with the arguments iuser and ruser as supplied to d03pdf/d03pda. You should use the arrays iuser and ruser to supply information to pdedef.

pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pdf/d03pda is called. Arguments denoted as Input must not be changed by this procedure.

Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pdf/d03pda. If your code inadvertently does return any NaNs or infinities, d03pdf/d03pda is likely to produce unexpected results.

6:     $\mathbf{bndary}$ – Subroutine, supplied by the user.External Procedure

bndary must compute the functions ${\beta }_i$ and ${\gamma }_i$ which define the boundary conditions as in equation (3).

The specification of bndary for d03pdf is:

Fortran Interface

Subroutine bndary ( npde, t, u, ux, ibnd, beta, gamma, ires) Integer, Intent (In) :: npde, ibnd Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde) Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)

C Header Interface

#include <nagmk26.h> void  bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *ibnd, double beta[], double gamma[], Integer *ires)

The specification of bndary for d03pda is:

Fortran Interface

Subroutine bndary ( npde, t, u, ux, ibnd, beta, gamma, ires, iuser, ruser) Integer, Intent (In) :: npde, ibnd Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)

C Header Interface

#include <nagmk26.h> void  bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *ibnd, double beta[], double gamma[], Integer *ires, Integer iuser[], double ruser[])

1:     $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

2:     $\mathbf{t}$ – Real (Kind=nag_wp)Input

On entry: the current value of the independent variable $t$.

3:     $\mathbf{u}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{u}\left(\mathit{i}\right)$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

4:     $\mathbf{ux}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{ux}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:     $\mathbf{ibnd}$ – IntegerInput

On entry: specifies which boundary conditions are to be evaluated.

$\mathbf{ibnd}=0$

bndary must set up the coefficients of the left-hand boundary, $x=a$.

$\mathbf{ibnd}\ne 0$

bndary must set up the coefficients of the right-hand boundary, $x=b$.

6:     $\mathbf{beta}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{beta}\left(\mathit{i}\right)$ must be set to the value of ${\beta }_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

7:     $\mathbf{gamma}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{gamma}\left(\mathit{i}\right)$ must be set to the value of ${\gamma }_{\mathit{i}}\left(x,t,U,U_x\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

8:     $\mathbf{ires}$ – IntegerInput/Output

On entry: set to $-1\text{ or }1$.

On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:

$\mathbf{ires}=2$

Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $\mathbf{ifail}=\mathbf{6}$.

$\mathbf{ires}=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $\mathbf{ires}=3$ when a physically meaningless input or output value has been generated. If you consecutively set $\mathbf{ires}=3$, d03pdf/d03pda returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.

9:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

10:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

bndary is called with the arguments iuser and ruser as supplied to d03pdf/d03pda. You should use the arrays iuser and ruser to supply information to bndary.

bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pdf/d03pda is called. Arguments denoted as Input must not be changed by this procedure.

Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pdf/d03pda. If your code inadvertently does return any NaNs or infinities, d03pdf/d03pda is likely to produce unexpected results.

7:     $\mathbf{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if $\mathbf{ind}=1$ the value of u must be unchanged from the previous call.

On exit: $\mathbf{u}\left(i,j\right)$ will contain the computed solution at $t=\mathbf{ts}$.

8:     $\mathbf{nbkpts}$ – IntegerInput

On entry: the number of break-points in the interval $\left[a,b\right]$.

Constraint: $\mathbf{nbkpts}\ge 2$.

9:     $\mathbf{xbkpts}\left(\mathbf{nbkpts}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the values of the break-points in the space direction. $\mathbf{xbkpts}\left(1\right)$ must specify the left-hand boundary, $a$, and $\mathbf{xbkpts}\left(\mathbf{nbkpts}\right)$ must specify the right-hand boundary, $b$.

Constraint: $\mathbf{xbkpts}\left(1\right)<\mathbf{xbkpts}\left(2\right)<\cdots <\mathbf{xbkpts}\left(\mathbf{nbkpts}\right)$.

10:   $\mathbf{npoly}$ – IntegerInput

On entry: the degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of break-points.

Constraint: $1\le \mathbf{npoly}\le 49$.

11:   $\mathbf{npts}$ – IntegerInput

On entry: the number of mesh points in the interval $\left[a,b\right]$.

Constraint: $\mathbf{npts}=\left(\mathbf{nbkpts}-1\right)\times \mathbf{npoly}+1$.

12:   $\mathbf{x}\left(\mathbf{npts}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the mesh points chosen by d03pdf/d03pda in the spatial direction. The values of x will satisfy $\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{npts}\right)$.

13:   $\mathbf{uinit}$ – Subroutine, supplied by the user.External Procedure

uinit must compute the initial values of the PDE components $U_{\mathit{i}}\left(x_{\mathit{j}},t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

The specification of uinit for d03pdf is:

Fortran Interface

Subroutine uinit ( npde, npts, x, u) Integer, Intent (In) :: npde, npts Real (Kind=nag_wp), Intent (In) :: x(npts) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts)

C Header Interface

#include <nagmk26.h> void  uinit (const Integer *npde, const Integer *npts, const double x[], double u[])

The specification of uinit for d03pda is:

Fortran Interface

Subroutine uinit ( npde, npts, x, u, iuser, ruser) Integer, Intent (In) :: npde, npts Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x(npts) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts)

C Header Interface

#include <nagmk26.h> void  uinit (const Integer *npde, const Integer *npts, const double x[], double u[], Integer iuser[], double ruser[])

1:     $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

2:     $\mathbf{npts}$ – IntegerInput

On entry: the number of mesh points in the interval $\left[a,b\right]$.

3:     $\mathbf{x}\left(\mathbf{npts}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{x}\left(\mathit{j}\right)$, contains the values of the $\mathit{j}$th mesh point, for $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

4:     $\mathbf{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ must be set to the initial value $U_{\mathit{i}}\left(x_{\mathit{j}},t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.

5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

uinit is called with the arguments iuser and ruser as supplied to d03pdf/d03pda. You should use the arrays iuser and ruser to supply information to uinit.

uinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pdf/d03pda is called. Arguments denoted as Input must not be changed by this procedure.

Note: uinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pdf/d03pda. If your code inadvertently does return any NaNs or infinities, d03pdf/d03pda is likely to produce unexpected results.

14:   $\mathbf{acc}$ – Real (Kind=nag_wp)Input

On entry: a positive quantity for controlling the local error estimate in the time integration. If $E\left(i,j\right)$ is the estimated error for $U_i$ at the $j$th mesh point, the error test is:

$$\left|E\left(i,j\right)\right|=\mathbf{acc}\times \left(1.0+\left|\mathbf{u}\left(i,j\right)\right|\right)\text{.}$$

Constraint: $\mathbf{acc}>0.0$.

15:   $\mathbf{rsave}\left(\mathbf{lrsave}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

If $\mathbf{ind}=0$, rsave need not be set on entry.

If $\mathbf{ind}=1$, rsave must be unchanged from the previous call to the routine because it contains required information about the iteration.

16:   $\mathbf{lrsave}$ – IntegerInput

On entry: the dimension of the array rsave as declared in the (sub)program from which d03pdf/d03pda is called.

Constraint: $\mathbf{lrsave}\ge 11\times \mathbf{npde}\times \mathbf{npts}+50+\mathit{nwkres}+\mathit{lenode}$.

17:   $\mathbf{isave}\left(\mathbf{lisave}\right)$ – Integer arrayCommunication Array

If $\mathbf{ind}=0$, isave need not be set on entry.

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the routine because it contains required information about the iteration. In particular:

$\mathbf{isave}\left(1\right)$

Contains the number of steps taken in time.

$\mathbf{isave}\left(2\right)$

Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.

$\mathbf{isave}\left(3\right)$

Contains the number of Jacobian evaluations performed by the time integrator.

$\mathbf{isave}\left(4\right)$

Contains the order of the last backward differentiation formula method used.

$\mathbf{isave}\left(5\right)$

Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.

18:   $\mathbf{lisave}$ – IntegerInput

On entry: the dimension of the array isave as declared in the (sub)program from which d03pdf/d03pda is called.

Constraint: $\mathbf{lisave}\ge \mathbf{npde}\times \mathbf{npts}+24$.

19:   $\mathbf{itask}$ – IntegerInput

On entry: specifies the task to be performed by the ODE integrator.

$\mathbf{itask}=1$

Normal computation of output values u at $t=\mathbf{tout}$.

$\mathbf{itask}=2$

One step and return.

$\mathbf{itask}=3$

Stop at first internal integration point at or beyond $t=\mathbf{tout}$.

Constraint: $\mathbf{itask}=1$, $2$ or $3$.

20:   $\mathbf{itrace}$ – IntegerInput

On entry: the level of trace information required from d03pdf/d03pda and the underlying ODE solver. itrace may take the value $-1$, $0$, $1$, $2$ or $3$.

$\mathbf{itrace}=-1$

No output is generated.

$\mathbf{itrace}=0$

Only warning messages from the PDE solver are printed on the current error message unit (see x04aaf).

$\mathbf{itrace}>0$

Output from the underlying ODE solver is printed on the current advisory message unit (see x04abf). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

If $\mathbf{itrace}<-1$, $-1$ is assumed and similarly if $\mathbf{itrace}>3$, $3$ is assumed.

The advisory messages are given in greater detail as itrace increases. You are advised to set $\mathbf{itrace}=0$, unless you are experienced with Sub-chapter D02M–N.

21:   $\mathbf{ind}$ – IntegerInput/Output

On entry: indicates whether this is a continuation call or a new integration.

$\mathbf{ind}=0$

Starts or restarts the integration in time.

$\mathbf{ind}=1$

Continues the integration after an earlier exit from the routine. In this case, only the arguments tout and ifail should be reset between calls to d03pdf/d03pda.

Constraint: $\mathbf{ind}=0$ or $1$.

On exit: $\mathbf{ind}=1$.

22:   $\mathbf{ifail}$ – IntegerInput/Output

Note: for d03pda, ifail does not occur in this position in the argument list. See the additional arguments described below.

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.

22:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

iuser is not used by d03pdf/d03pda, but is passed directly to pdedef, bndary and uinit and may be used to pass information to these routines.

23:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

24:   $\mathbf{cwsav}\left(10\right)$ – Character(80) arrayCommunication Array

25:   $\mathbf{lwsav}\left(100\right)$ – Logical arrayCommunication Array

26:   $\mathbf{iwsav}\left(505\right)$ – Integer arrayCommunication Array

27:   $\mathbf{rwsav}\left(1100\right)$ – Real (Kind=nag_wp) arrayCommunication Array

28:   $\mathbf{ifail}$ – IntegerInput/Output

Note: see the argument description for ifail above.

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{acc}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{acc}>0.0$.

On entry, $\mathit{i}=〈\mathit{\text{value}}〉$, $\mathbf{xbkpts}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$, $\mathit{j}=〈\mathit{\text{value}}〉$ and $\mathbf{xbkpts}\left(\mathit{j}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xbkpts}\left(1\right)<\mathbf{xbkpts}\left(2\right)<\cdots <\mathbf{xbkpts}\left(\mathbf{nbkpts}\right)$.

On entry, $\mathbf{ind}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ind}=0$ or $1$.

On entry, $\mathbf{itask}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itask}=1$, $2$ or $3$.

On entry, $\mathbf{lisave}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lisave}\ge 〈\mathit{\text{value}}〉$.

On entry, $\mathbf{lrsave}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lrsave}\ge 〈\mathit{\text{value}}〉$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}=0$, $1$ or $2$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{xbkpts}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\le 0$ or $\mathbf{xbkpts}\left(1\right)\ge 0.0$ 

On entry, $\mathbf{nbkpts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nbkpts}\ge 2$.

On entry, $\mathbf{npde}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npde}\ge 1$.

On entry, $\mathbf{npoly}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npoly}\le 49$.

On entry, $\mathbf{npoly}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npoly}\ge 1$.

On entry, $\mathbf{npts}=〈\mathit{\text{value}}〉$, $\mathbf{nbkpts}=〈\mathit{\text{value}}〉$ and $\mathbf{npoly}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npts}=\left(\mathbf{nbkpts}-1\right)\times \mathbf{npoly}+1$.

On entry, on initial entry $\mathbf{ind}=1$.
Constraint: on initial entry $\mathbf{ind}=0$.

On entry, $\mathbf{tout}=〈\mathit{\text{value}}〉$ and $\mathbf{ts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{tout}>\mathbf{ts}$.

On entry, $\mathbf{tout}-\mathbf{ts}$ is too small: $\mathbf{tout}=〈\mathit{\text{value}}〉$ and $\mathbf{ts}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=2$

Underlying ODE solver cannot make further progress from the point ts with the supplied value of acc. $\mathbf{ts}=〈\mathit{\text{value}}〉$, $\mathbf{acc}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=3$

Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts: $\mathbf{ts}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=4$

In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting $\mathbf{ires}=3$ in pdedef or bndary.

$\mathbf{ifail}=5$

Singular Jacobian of ODE system. Check problem formulation.

$\mathbf{ifail}=6$

In evaluating residual of ODE system, $\mathbf{ires}=2$ has been set in pdedef or bndary. Integration is successful as far as ts: $\mathbf{ts}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=7$

acc was too small to start integration: $\mathbf{acc}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=8$

ires set to an invalid value in call to pdedef or bndary.

$\mathbf{ifail}=9$

Serious error in internal call to an auxiliary. Increase itrace for further details.

$\mathbf{ifail}=10$

Integration completed, but a small change in acc is unlikely to result in a changed solution. $\mathbf{acc}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=11$

Error during Jacobian formulation for ODE system. Increase itrace for further details.

$\mathbf{ifail}=14$

Flux function appears to depend on time derivatives.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d03pdfe.f90)

Program Text (d03pdae.f90)

10.2

Program Data

Program Data (d03pdfe.d)

Program Data (d03pdae.d)

10.3

Program Results

Program Results (d03pdfe.r)

Program Results (d03pdae.r)