NAG Library Routine Document

d03ppf  (dim1_parab_remesh_fd_old)
d03ppa (dim1_parab_remesh_fd)

1

Purpose

2

Specification

2.1

Specification for d03ppf

2.2

Specification for d03ppa

3

Description

4

References

5

Arguments

1:     $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs 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 evaluate the functions $P_{i,j}$, $Q_i$ and $R_i$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$, $U_x$ and $V$. $Q_i$ may depend linearly on $\stackrel{.}{V}$. pdedef is called approximately midway between each pair of mesh points in turn by d03ppf/d03ppa.

The specification of pdedef for d03ppf is:

Fortran Interface

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

C Header Interface

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

The specification of pdedef for d03ppa is:

Fortran Interface

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

C Header Interface

#include <nagmk26.h> void  pdedef (const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], 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:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the current value of the space variable $x$.

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

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

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

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

6:     $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

7:     $\mathbf{v}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

8:     $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly in $Q_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

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

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

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

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

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

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

12:   $\mathbf{ires}$ – 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 $\mathbf{ires}=3$, d03ppf/d03ppa 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 d03ppa. Users of d03ppf therefore need not read the remainder of this description.

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

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

pdedef is called with the arguments iuser and ruser as supplied to d03ppf/d03ppa. 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 d03ppf/d03ppa 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 d03ppf/d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/d03ppa is likely to produce unexpected results.

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

bndary must evaluate the functions ${\beta }_i$ and ${\gamma }_i$ which describe the boundary conditions, as given in (4).

The specification of bndary for d03ppf is:

Fortran Interface

Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, gamma, ires) Integer, Intent (In) :: npde, nv, ibnd Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv) 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 *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires)

The specification of bndary for d03ppa is:

Fortran Interface

Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, gamma, ires, iuser, ruser) Integer, Intent (In) :: npde, nv, ibnd Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv) 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 *nv, const double v[], const double vdot[], 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:     $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

6:     $\mathbf{v}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

7:     $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{vdot}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly in ${\gamma }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

8:     $\mathbf{ibnd}$ – 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, $x=b$.

9:     $\mathbf{beta}\left(\mathbf{npde}\right)$ – 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 $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

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

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

11:   $\mathbf{ires}$ – 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 $\mathbf{ires}=3$, d03ppf/d03ppa 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 d03ppa. Users of d03ppf therefore need not read the remainder of this description.

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

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

bndary is called with the arguments iuser and ruser as supplied to d03ppf/d03ppa. 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 d03ppf/d03ppa 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 d03ppf/d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/d03ppa is likely to produce unexpected results.

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

uvinit must supply the initial $\left(t=t_0\right)$ values of $U\left(x,t\right)$ and $V\left(t\right)$ for all values of $x$ in the interval $a\le x\le b$.

The specification of uvinit for d03ppf is:

Fortran Interface

Subroutine uvinit ( npde, npts, nxi, x, xi, u, nv, v) Integer, Intent (In) :: npde, npts, nxi, nv Real (Kind=nag_wp), Intent (In) :: x(npts), xi(nxi) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts), v(nv)

C Header Interface

#include <nagmk26.h> void  uvinit (const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[])

The specification of uvinit for d03ppa is:

Fortran Interface

Subroutine uvinit ( npde, npts, nxi, x, xi, u, nv, v, iuser, ruser) Integer, Intent (In) :: npde, npts, nxi, nv Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x(npts), xi(nxi) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts), v(nv)

C Header Interface

#include <nagmk26.h> void  uvinit (const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[], 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:     $\mathbf{nxi}$ – IntegerInput

On entry: the number of ODE/PDE coupling points.

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

On entry: the current mesh. $\mathbf{x}\left(\mathit{i}\right)$ contains the value of $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npts}$.

5:     $\mathbf{xi}\left(\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left(\mathit{i}\right)$ contains the value of the ODE/PDE coupling point, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$.

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

On exit: $\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component $U_{\mathit{i}}\left(x_{\mathit{j}},t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

7:     $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

8:     $\mathbf{v}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{v}\left(\mathit{i}\right)$ contains the value of component $V_{\mathit{i}}\left(t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf 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

uvinit is called with the arguments iuser and ruser as supplied to d03ppf/d03ppa. You should use the arrays iuser and ruser to supply information to uvinit.

uvinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03ppf/d03ppa is called. Arguments denoted as Input must not be changed by this procedure.

Note: uvinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03ppf/d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/d03ppa is likely to produce unexpected results.

8:     $\mathbf{u}\left(\mathbf{neqn}\right)$ – 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: $\mathbf{u}\left(\mathbf{npde}\times \left(\mathit{j}-1\right)+\mathit{i}\right)$ contains the computed solution $U_{\mathit{i}}\left(x_{\mathit{j}},t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and $\mathbf{u}\left(\mathbf{npts}\times \mathbf{npde}+\mathit{k}\right)$ contains $V_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,\mathbf{nv}$, evaluated at $t=\mathbf{ts}$.

9:     $\mathbf{npts}$ – IntegerInput

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

Constraint: $\mathbf{npts}\ge 3$.

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

On entry: the initial mesh points in the space direction. $\mathbf{x}\left(1\right)$ must specify the left-hand boundary, $a$, and $\mathbf{x}\left(\mathbf{npts}\right)$ must specify the right-hand boundary, $b$.

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

On exit: the final values of the mesh points.

11:   $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODE in the system.

Constraint: $\mathbf{nv}\ge 0$.

12:   $\mathbf{odedef}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

odedef must evaluate the functions $F$, which define the system of ODEs, as given in (3).

If you wish to compute the solution of a system of PDEs only ($\mathbf{nv}=0$), odedef must be the dummy routine d03pck for d03ppf (or d53pck for d03ppa). d03pck and d53pck are included in the NAG Library.

The specification of odedef for d03ppf is:

Fortran Interface

Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, f, ires) Integer, Intent (In) :: npde, nv, nxi Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi) Real (Kind=nag_wp), Intent (Out) :: f(nv)

C Header Interface

#include <nagmk26.h> void  odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires)

The specification of odedef for d03ppa is:

Fortran Interface

Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, f, ires, iuser, ruser) Integer, Intent (In) :: npde, nv, nxi Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: f(nv)

C Header Interface

#include <nagmk26.h> void  odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], 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:     $\mathbf{nv}$ – IntegerInput

On entry: the number of coupled ODEs in the system.

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

On entry: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ contains the value of the component $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

5:     $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nv}>0$, $\mathbf{vdot}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

6:     $\mathbf{nxi}$ – IntegerInput

On entry: the number of ODE/PDE coupling points.

7:     $\mathbf{xi}\left(\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left(\mathit{i}\right)$ contains the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$.

8:     $\mathbf{ucp}\left(\mathbf{npde},\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucp}\left(\mathit{i},\mathit{j}\right)$ contains the value of $U_{\mathit{i}}\left(x,t\right)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

9:     $\mathbf{ucpx}\left(\mathbf{npde},\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpx}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

10:   $\mathbf{rcp}\left(\mathbf{npde},\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{rcp}\left(\mathit{i},\mathit{j}\right)$ contains the value of the flux $R_{\mathit{i}}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

11:   $\mathbf{ucpt}\left(\mathbf{npde},\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpt}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial U_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

12:   $\mathbf{ucptx}\left(\mathbf{npde},\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{ucptx}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{{\partial }^2{U}_{\mathit{i}}}{\partial x\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

13:   $\mathbf{f}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{f}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $F$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, where $F$ is defined as

$$F=G-A\stackrel{.}{V}-B\left(\begin{array}{c}{U}_t^{*}\\ U_{xt}^{*}\end{array}\right)\text{,}$$

(5)

or

$$F=-A\stackrel{.}{V}-B\left(\begin{array}{c}{U}_t^{*}\\ U_{xt}^{*}\end{array}\right)\text{.}$$

(6)

The definition of $F$ is determined by the input value of ires.

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

On entry: the form of $F$ that must be returned in the array f.

$\mathbf{ires}=1$

Equation (5) must be used.

$\mathbf{ires}=-1$

Equation (6) must be used.

On exit: should usually remain unchanged. However, you may reset 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 $\mathbf{ires}=3$, d03ppf/d03ppa 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 d03ppa. Users of d03ppf therefore need not read the remainder of this description.

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

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

odedef is called with the arguments iuser and ruser as supplied to d03ppf/d03ppa. You should use the arrays iuser and ruser to supply information to odedef.

odedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03ppf/d03ppa is called. Arguments denoted as Input must not be changed by this procedure.

Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03ppf/d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/d03ppa is likely to produce unexpected results.

13:   $\mathbf{nxi}$ – IntegerInput

On entry: the number of ODE/PDE coupling points.

Constraints:

• if $\mathbf{nv}=0$, $\mathbf{nxi}=0$;
• if $\mathbf{nv}>0$, $\mathbf{nxi}\ge 0$.

14:   $\mathbf{xi}\left(\mathbf{nxi}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: if $\mathbf{nxi}>0$, $\mathbf{xi}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nxi}$, must be set to the ODE/PDE coupling points.

Constraint: $\mathbf{x}\left(1\right)\le \mathbf{xi}\left(1\right)<\mathbf{xi}\left(2\right)<\cdots <\mathbf{xi}\left(\mathbf{nxi}\right)\le \mathbf{x}\left(\mathbf{npts}\right)$.

15:   $\mathbf{neqn}$ – IntegerInput

On entry: the number of ODEs in the time direction.

Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

16:   $\mathbf{rtol}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array rtol must be at least $1$ if $\mathbf{itol}=1$ or $2$ and at least $\mathbf{neqn}$ if $\mathbf{itol}=3$ or $4$.

On entry: the relative local error tolerance.

Constraint: $\mathbf{rtol}\left(i\right)\ge 0.0$ for all relevant $i$.

17:   $\mathbf{atol}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array atol must be at least $1$ if $\mathbf{itol}=1$ or $3$ and at least $\mathbf{neqn}$ if $\mathbf{itol}=2$ or $4$.

On entry: the absolute local error tolerance.

Constraints:

• $\mathbf{atol}\left(i\right)\ge 0.0$ for all relevant $i$;
• Corresponding elements of atol and rtol cannot both be $0.0$.

18:   $\mathbf{itol}$ – IntegerInput

On entry: a value to indicate the form of the local error test. itol indicates to d03ppf/d03ppa whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $‖e_i/w_i‖<1.0$, where $w_i$ is defined as follows:

itol	rtol	atol	$w_i$
1	scalar	scalar	$\mathbf{rtol}\left(1\right)\times \left|U_i\right|+\mathbf{atol}\left(1\right)$
2	scalar	vector	$\mathbf{rtol}\left(1\right)\times \left|U_i\right|+\mathbf{atol}\left(i\right)$
3	vector	scalar	$\mathbf{rtol}\left(i\right)\times \left|U_i\right|+\mathbf{atol}\left(1\right)$
4	vector	vector	$\mathbf{rtol}\left(i\right)\times \left|U_i\right|+\mathbf{atol}\left(i\right)$

In the above, $e_{\mathit{i}}$ denotes the estimated local error for the $\mathit{i}$th component of the coupled PDE/ODE system in time, $\mathbf{u}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{neqn}$.

The choice of norm used is defined by the argument norm.

Constraint: $1\le \mathbf{itol}\le 4$.

19:   $\mathbf{norm}$ – Character(1)Input

On entry: the type of norm to be used.

$\mathbf{norm}=\text{'M'}$

Maximum norm.

$\mathbf{norm}=\text{'A'}$

Averaged $L_2$ norm.

If ${\mathbf{u}}_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged $L_2$ norm

$${\mathbf{u}}_{\mathrm{norm}}=\sqrt{\frac{1}{\mathbf{neqn}}\sum _{i=1}^{\mathbf{neqn}}{\left(\mathbf{u}\left(i\right)/w_i\right)}^2}\text{,}$$

while for the maximum norm

$${\mathbf{u}}_{\mathrm{norm}}=\underset{i}{\max }\left|\mathbf{u}\left(i\right)/w_i\right|\text{.}$$

See the description of itol for the formulation of the weight vector $w$.

Constraint: $\mathbf{norm}=\text{'M'}$ or $\text{'A'}$.

20:   $\mathbf{laopt}$ – Character(1)Input

On entry: the type of matrix algebra required.

$\mathbf{laopt}=\text{'F'}$

Full matrix methods to be used.

$\mathbf{laopt}=\text{'B'}$

Banded matrix methods to be used.

$\mathbf{laopt}=\text{'S'}$

Sparse matrix methods to be used.

Constraint: $\mathbf{laopt}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.

Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., $\mathbf{nv}=0$).

21:   $\mathbf{algopt}\left(30\right)$ – Real (Kind=nag_wp) arrayInput

On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, $\mathbf{algopt}\left(1\right)$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:

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

Selects the ODE integration method to be used. If $\mathbf{algopt}\left(1\right)=1.0$, a BDF method is used and if $\mathbf{algopt}\left(1\right)=2.0$, a Theta method is used. The default value is $\mathbf{algopt}\left(1\right)=1.0$.

If $\mathbf{algopt}\left(1\right)=2.0$, $\mathbf{algopt}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,4$ are not used.

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

Specifies the maximum order of the BDF integration formula to be used. $\mathbf{algopt}\left(2\right)$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is $\mathbf{algopt}\left(2\right)=5.0$.

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

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If $\mathbf{algopt}\left(3\right)=1.0$ a modified Newton iteration is used and if $\mathbf{algopt}\left(3\right)=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, there is an automatic switch to the modified Newton iteration. The default value is $\mathbf{algopt}\left(3\right)=1.0$.

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

Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as $P_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$, for some $i$ or when there is no ${\stackrel{.}{V}}_i\left(t\right)$ dependence in the coupled ODE system. If $\mathbf{algopt}\left(4\right)=1.0$, the Petzold test is used. If $\mathbf{algopt}\left(4\right)=2.0$, the Petzold test is not used. The default value is $\mathbf{algopt}\left(4\right)=1.0$.

If $\mathbf{algopt}\left(1\right)=1.0$, $\mathbf{algopt}\left(\mathit{i}\right)$, for $\mathit{i}=5,6,7$, are not used.

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

Specifies the value of Theta to be used in the Theta integration method. $0.51\le \mathbf{algopt}\left(5\right)\le 0.99$. The default value is $\mathbf{algopt}\left(5\right)=0.55$.

$\mathbf{algopt}\left(6\right)$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If $\mathbf{algopt}\left(6\right)=1.0$, a modified Newton iteration is used and if $\mathbf{algopt}\left(6\right)=2.0$, a functional iteration method is used. The default value is $\mathbf{algopt}\left(6\right)=1.0$.

$\mathbf{algopt}\left(7\right)$

Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If $\mathbf{algopt}\left(7\right)=1.0$, switching is allowed and if $\mathbf{algopt}\left(7\right)=2.0$, switching is not allowed. The default value is $\mathbf{algopt}\left(7\right)=1.0$.

$\mathbf{algopt}\left(11\right)$

Specifies a point in the time direction, $t_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of $t_{\mathrm{crit}}$ is described under the argument itask. If $\mathbf{algopt}\left(1\right)\ne 0.0$, a value of $0.0$ for $\mathbf{algopt}\left(11\right)$, say, should be specified even if itask subsequently specifies that $t_{\mathrm{crit}}$ will not be used.

$\mathbf{algopt}\left(12\right)$

Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, $\mathbf{algopt}\left(12\right)$ should be set to $0.0$.

$\mathbf{algopt}\left(13\right)$

Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, $\mathbf{algopt}\left(13\right)$ should be set to $0.0$.

$\mathbf{algopt}\left(14\right)$

Specifies the initial step size to be attempted by the integrator. If $\mathbf{algopt}\left(14\right)=0.0$, the initial step size is calculated internally.

$\mathbf{algopt}\left(15\right)$

Specifies the maximum number of steps to be attempted by the integrator in any one call. If $\mathbf{algopt}\left(15\right)=0.0$, no limit is imposed.

$\mathbf{algopt}\left(23\right)$

Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, $U_t$, $V$ and $\stackrel{.}{V}$. If $\mathbf{algopt}\left(23\right)=1.0$, a modified Newton iteration is used and if $\mathbf{algopt}\left(23\right)=2.0$, functional iteration is used. The default value is $\mathbf{algopt}\left(23\right)=1.0$.

$\mathbf{algopt}\left(29\right)$ and $\mathbf{algopt}\left(30\right)$ are used only for the sparse matrix algebra option, $\mathbf{laopt}=\text{'S'}$.

$\mathbf{algopt}\left(29\right)$

Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<\mathbf{algopt}\left(29\right)<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If $\mathbf{algopt}\left(29\right)$ lies outside this range then the default value is used. If the routines regard the Jacobian matrix as numerically singular then increasing $\mathbf{algopt}\left(29\right)$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is $\mathbf{algopt}\left(29\right)=0.1$.

$\mathbf{algopt}\left(30\right)$

Is used as a relative pivot threshold during subsequent Jacobian decompositions (see $\mathbf{algopt}\left(29\right)$) below which an internal error is invoked. If $\mathbf{algopt}\left(30\right)$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see $\mathbf{algopt}\left(29\right)$). The default value is $\mathbf{algopt}\left(30\right)=0.0001$.

22:   $\mathbf{remesh}$ – LogicalInput

On entry: indicates whether or not spatial remeshing should be performed.

$\mathbf{remesh}=\mathrm{.TRUE.}$

Indicates that spatial remeshing should be performed as specified.

$\mathbf{remesh}=\mathrm{.FALSE.}$

Indicates that spatial remeshing should be suppressed.

Note: remesh should not be changed between consecutive calls to d03ppf/d03ppa. Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.

23:   $\mathbf{nxfix}$ – IntegerInput

On entry: the number of fixed mesh points.

Constraint: $0\le \mathbf{nxfix}\le \mathbf{npts}-2$.

Note: the end points $\mathbf{x}\left(1\right)$ and $\mathbf{x}\left(\mathbf{npts}\right)$ are fixed automatically and hence should not be specified as fixed points.

24:   $\mathbf{xfix}\left(\mathbf{nxfix}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{xfix}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nxfix}$, must contain the value of the $x$ coordinate at the $\mathit{i}$th fixed mesh point.

Constraints:

• $\mathbf{xfix}\left(\mathit{i}\right)<\mathbf{xfix}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nxfix}-1$;
• each fixed mesh point must coincide with a user-supplied initial mesh point, that is $\mathbf{xfix}\left(\mathit{i}\right)=\mathbf{x}\left(\mathit{j}\right)$ for some $j$, $2\le j\le \mathbf{npts}-1$.

Note: the positions of the fixed mesh points in the array x remain fixed during remeshing, and so the number of mesh points between adjacent fixed points (or between fixed points and end points) does not change. You should take this into account when choosing the initial mesh distribution.

25:   $\mathbf{nrmesh}$ – IntegerInput

On entry: specifies the spatial remeshing frequency and criteria for the calculation and adoption of a new mesh.

$\mathbf{nrmesh}<0$

Indicates that a new mesh is adopted according to the argument dxmesh. The mesh is tested every $\left|\mathbf{nrmesh}\right|$ timesteps.

$\mathbf{nrmesh}=0$

Indicates that remeshing should take place just once at the end of the first time step reached when $t>\mathbf{trmesh}$.

$\mathbf{nrmesh}>0$

Indicates that remeshing will take place every nrmesh time steps, with no testing using dxmesh.

Note: nrmesh may be changed between consecutive calls to d03ppf/d03ppa to give greater flexibility over the times of remeshing.

26:   $\mathbf{dxmesh}$ – Real (Kind=nag_wp)Input

On entry: determines whether a new mesh is adopted when nrmesh is set less than zero. A possible new mesh is calculated at the end of every $\left|\mathbf{nrmesh}\right|$ time steps, but is adopted only if

$$x_i^{\left(\mathrm{new}\right)}>x_i^{\left(\mathrm{old}\right)}+\mathbf{dxmesh}\times \left(x_{i+1}^{\left(\mathrm{old}\right)}-x_i^{\left(\mathrm{old}\right)}\right)$$

or

$$x_i^{\left(\mathrm{new}\right)}<x_i^{\left(\mathrm{old}\right)}-\mathbf{dxmesh}\times \left(x_i^{\left(\mathrm{old}\right)}-x_{i-1}^{\left(\mathrm{old}\right)}\right)$$

dxmesh thus imposes a lower limit on the difference between one mesh and the next.

Constraint: $\mathbf{dxmesh}\ge 0.0$.

27:   $\mathbf{trmesh}$ – Real (Kind=nag_wp)Input

On entry: specifies when remeshing will take place when nrmesh is set to zero. Remeshing will occur just once at the end of the first time step reached when $t$ is greater than trmesh.

Note: trmesh may be changed between consecutive calls to d03ppf/d03ppa to force remeshing at several specified times.

28:   $\mathbf{ipminf}$ – IntegerInput

On entry: the level of trace information regarding the adaptive remeshing. Details are directed to the current advisory message unit (see x04abf).

$\mathbf{ipminf}=0$

No trace information.

$\mathbf{ipminf}=1$

Brief summary of mesh characteristics.

$\mathbf{ipminf}=2$

More detailed information, including old and new mesh points, mesh sizes and monitor function values.

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

29:   $\mathbf{xratio}$ – Real (Kind=nag_wp)Input

On entry: an input bound on the adjacent mesh ratio (greater than $1.0$ and typically in the range $1.5$ to $3.0$). The remeshing routines will attempt to ensure that

$$\left(x_i-x_{i-1}\right)/\mathbf{xratio}<x_{i+1}-x_i<\mathbf{xratio}\times \left(x_i-x_{i-1}\right)\text{.}$$

Suggested value: $\mathbf{xratio}=1.5$.

Constraint: $\mathbf{xratio}>1.0$.

30:   $\mathbf{con}$ – Real (Kind=nag_wp)Input

On entry: an input bound on the sub-integral of the monitor function $F^{\mathrm{mon}}\left(x\right)$ over each space step. The remeshing routines will attempt to ensure that

$$\underset{x_i}{\overset{x_{i+1}}{\int }}{F}^{\mathrm{mon}}\left(x\right)dx\le \mathbf{con}\underset{x_1}{\overset{x_{\mathbf{npts}}}{\int }}{F}^{\mathrm{mon}}\left(x\right)dx\text{,}$$

(see Furzeland (1984)). con gives you more control over the mesh distribution e.g., decreasing con allows more clustering. A typical value is $2/\left(\mathbf{npts}-1\right)$, but you are encouraged to experiment with different values. Its value is not critical and the mesh should be qualitatively correct for all values in the range given below.

Suggested value: $\mathbf{con}=2.0/\left(\mathbf{npts}-1\right)$.

Constraint: $0.1/\left(\mathbf{npts}-1\right)\le \mathbf{con}\le 10.0/\left(\mathbf{npts}-1\right)$.

31:   $\mathbf{monitf}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

monitf must supply and evaluate a remesh monitor function to indicate the solution behaviour of interest.

If you specify $\mathbf{remesh}=\mathrm{.FALSE.}$, i.e., no remeshing, monitf will not be called and the dummy routine d03pcl for d03ppf (or d53pcl for d03ppa) may be used for monitf. (d03pcl and d53pcl are included in the NAG Library.)

The specification of monitf for d03ppf is:

Fortran Interface

Subroutine monitf ( t, npts, npde, x, u, r, fmon) Integer, Intent (In) :: npts, npde Real (Kind=nag_wp), Intent (In) :: t, x(npts), u(npde,npts), r(npde,npts) Real (Kind=nag_wp), Intent (Out) :: fmon(npts)

C Header Interface

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

The specification of monitf for d03ppa is:

Fortran Interface

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

C Header Interface

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

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

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

2:     $\mathbf{npts}$ – IntegerInput

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

3:     $\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.

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

On entry: the current mesh. $\mathbf{x}\left(\mathit{i}\right)$ contains the value of $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npts}$.

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

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

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

On entry: $\mathbf{r}\left(\mathit{i},\mathit{j}\right)$ contains the value of $R_{\mathit{i}}\left(x,t,U,U_x,V\right)$ at $x=\mathbf{x}\left(\mathit{j}\right)$ and time $t$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

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

On exit: $\mathbf{fmon}\left(i\right)$ must contain the value of the monitor function $F^{\mathrm{mon}}\left(x\right)$ at mesh point $x=\mathbf{x}\left(i\right)$.

Constraint: $\mathbf{fmon}\left(i\right)\ge 0.0$.

Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf therefore need not read the remainder of this description.

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

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

monitf is called with the arguments iuser and ruser as supplied to d03ppf/d03ppa. You should use the arrays iuser and ruser to supply information to monitf.

monitf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03ppf/d03ppa is called. Arguments denoted as Input must not be changed by this procedure.

Note: monitf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03ppf/d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/d03ppa is likely to produce unexpected results.

32:   $\mathbf{rsave}\left(\mathbf{lrsave}\right)$ – 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.

33:   $\mathbf{lrsave}$ – IntegerInput

On entry: the dimension of the array rsave as declared in the (sub)program from which d03ppf/d03ppa is called. Its size depends on the type of matrix algebra selected.

If $\mathbf{laopt}=\text{'F'}$, $\mathbf{lrsave}\ge \mathbf{neqn}\times \mathbf{neqn}+\mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

If $\mathbf{laopt}=\text{'B'}$, $\mathbf{lrsave}\ge \left(3\times \mathit{mlu}+1\right)\times \mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

If $\mathbf{laopt}=\text{'S'}$, $\mathbf{lrsave}\ge 4\times \mathbf{neqn}+11\times \mathbf{neqn}/2+1+\mathit{nwkres}+\mathit{lenode}$.

Where

	$\mathit{mlu}$ is the lower or upper half bandwidths such that $\mathit{mlu}=2\times \mathbf{npde}-1$, for PDE problems only; or $\mathit{mlu}=\mathbf{neqn}-1$, for coupled PDE/ODE problems.
	$\mathit{nwkres}=\left\{\begin{array}{ll}\mathbf{npde}\times \left(3\times \mathbf{npde}+6\times \mathbf{nxi}+\mathbf{npts}+15\right)+\mathbf{nxi}+\mathbf{nv}+7\times \mathbf{npts}+\mathbf{nxfix}+1\text{,}& \text{when }\mathbf{nv}>0\text{ and }\mathbf{nxi}>0\text{;}\\ \mathbf{npde}\times \left(3\times \mathbf{npde}+\mathbf{npts}+21\right)+\mathbf{nv}+7\times \mathbf{npts}+\mathbf{nxfix}+2\text{,}& \text{when }\mathbf{nv}>0\text{ and }\mathbf{nxi}=0\text{; or}\\ \mathbf{npde}\times \left(3\times \mathbf{npde}+\mathbf{npts}+21\right)+7\times \mathbf{npts}+\mathbf{nxfix}+3\text{,}& \text{when }\mathbf{nv}=0\text{.}\end{array}\right.$
	$\mathit{lenode}=\left\{\begin{array}{cc}\left(6+\mathrm{int}\left(\mathbf{algopt}\left(2\right)\right)\right)\times \mathbf{neqn}+50\text{,}& \text{when the BDF method is used;}\\ 9\times \mathbf{neqn}+50\text{,}& \text{when the Theta method is used.}\end{array}\right.$

Note: when using the sparse option, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if $\mathbf{itrace}>0$ and the routine returns with $\mathbf{ifail}=\mathbf{15}$.

34:   $\mathbf{isave}\left(\mathbf{lisave}\right)$ – 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 required for subsequent calls. 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 ODE method last used in the time integration.

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

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

The rest of the array is used as workspace.

35:   $\mathbf{lisave}$ – IntegerInput

On entry: the dimension of the array isave as declared in the (sub)program from which d03ppf/d03ppa is called.

Its size depends on the type of matrix algebra selected:

• if $\mathbf{laopt}=\text{'B'}$, $\mathbf{lisave}\ge \mathbf{neqn}+25+\mathbf{nxfix}$;
• if $\mathbf{laopt}=\text{'F'}$, $\mathbf{lisave}\ge 25+\mathbf{nxfix}$;
• if $\mathbf{laopt}=\text{'S'}$, $\mathbf{lisave}\ge 25\times \mathbf{neqn}+25+\mathbf{nxfix}$.

Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed on the current error message unit if $\mathbf{itrace}>0$ and the routine returns with $\mathbf{ifail}=\mathbf{15}$.

36:   $\mathbf{itask}$ – 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 $t=\mathbf{tout}$.

$\mathbf{itask}=4$

Normal computation of output values u at $t=\mathbf{tout}$ but without overshooting $t=t_{\mathrm{crit}}$ where $t_{\mathrm{crit}}$ is described under the argument algopt.

$\mathbf{itask}=5$

Take one step in the time direction and return, without passing $t_{\mathrm{crit}}$, where $t_{\mathrm{crit}}$ is described under the argument algopt.

Constraint: $\mathbf{itask}=1$, $2$, $3$, $4$ or $5$.

37:   $\mathbf{itrace}$ – IntegerInput

On entry: the level of trace information required from d03ppf/d03ppa and the underlying ODE solver:

$\mathbf{itrace}\le -1$

No output is generated.

$\mathbf{itrace}=0$

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

$\mathbf{itrace}=1$

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.

$\mathbf{itrace}=2$

Output from the underlying ODE solver is similar to that produced when $\mathbf{itrace}=1$, except that the advisory messages are given in greater detail.

$\mathbf{itrace}\ge 3$

Output from the underlying ODE solver is similar to that produced when $\mathbf{itrace}=2$, except that the advisory messages are given in greater detail.

You are advised to set $\mathbf{itrace}=0$, unless you are experienced with Sub-chapter D02M–N.

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

On entry: must be set to $0$ or $1$.

$\mathbf{ind}=0$

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 and the remeshing arguments nrmesh, dxmesh, trmesh, xratio and con may be reset between calls to d03ppf/d03ppa.

Constraint: $0\le \mathbf{ind}\le 1$.

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

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

Note: for d03ppa, 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 d03ppa. Users of d03ppf therefore need not read the remainder of this description.

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

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

iuser and ruser are not used by d03ppf/d03ppa, but are passed directly to pdedef, bndary, uvinit, odedef and monitf and may be used to pass information to these routines.

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

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

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

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

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

Note: see the argument description for ifail above.

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, at least one point in xi lies outside $\left[\mathbf{x}\left(1\right),\mathbf{x}\left(\mathbf{npts}\right)\right]$: $\mathbf{x}\left(1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left(\mathbf{npts}\right)=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{con}=〈\mathit{\text{value}}〉$, $\mathbf{npts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{con}\le 10.0/\left(\mathbf{npts}-1\right)$.

On entry, $\mathbf{con}=〈\mathit{\text{value}}〉$, $\mathbf{npts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{con}\ge 0.1/\left(\mathbf{npts}-1\right)$.

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

On entry, $\mathit{i}=〈\mathit{\text{value}}〉$, $\mathbf{xfix}\left(\mathit{i}+1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{xfix}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xfix}\left(\mathit{i}+1\right)>\mathbf{xfix}\left(\mathit{i}\right)$.

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

On entry, $\mathit{i}=〈\mathit{\text{value}}〉$, $\mathbf{xi}\left(\mathit{i}+1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{xi}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xi}\left(\mathit{i}+1\right)>\mathbf{xi}\left(\mathit{i}\right)$.

On entry, $\mathit{i}=〈\mathit{\text{value}}〉$ and $\mathbf{atol}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{atol}\left(\mathit{i}\right)\ge 0.0$.

On entry, $\mathit{i}=〈\mathit{\text{value}}〉$ and $\mathit{j}=〈\mathit{\text{value}}〉$.
Constraint: corresponding elements $\mathbf{atol}\left(\mathit{i}\right)$ and $\mathbf{rtol}\left(\mathit{j}\right)$ cannot both be $0.0$.

On entry, $\mathit{i}=〈\mathit{\text{value}}〉$ and $\mathbf{rtol}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{rtol}\left(\mathit{i}\right)\ge 0.0$.

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

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

On entry, $\mathbf{itask}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itask}=1$, $2$, $3$, $4$ or $5$.

On entry, $\mathbf{itol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itol}=1$, $2$, $3$ or $4$.

On entry, $\mathbf{laopt}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{laopt}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.

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

On entry, $\mathbf{neqn}=〈\mathit{\text{value}}〉$, $\mathbf{npde}=〈\mathit{\text{value}}〉$, $\mathbf{npts}=〈\mathit{\text{value}}〉$ and $\mathbf{nv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

On entry, $\mathbf{norm}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{norm}=\text{'A'}$ or $\text{'M'}$.

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

On entry, $\mathbf{npts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{npts}\ge 3$.

On entry, $\mathbf{nv}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nv}\ge 0$.

On entry, $\mathbf{nv}=〈\mathit{\text{value}}〉$ and $\mathbf{nxi}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxi}=0$ when $\mathbf{nv}=0$.

On entry, $\mathbf{nv}=〈\mathit{\text{value}}〉$ and $\mathbf{nxi}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxi}\ge 0$ when $\mathbf{nv}>0$.

On entry, $\mathbf{nxfix}=〈\mathit{\text{value}}〉$, $\mathbf{npts}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxfix}\le \mathbf{npts}-2$.

On entry, $\mathbf{nxfix}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxfix}\ge 0$.

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

On entry, the point $\mathbf{xfix}\left(\mathit{i}\right)$ does not coincide with any $\mathbf{x}\left(\mathit{j}\right)$: $\mathit{i}=〈\mathit{\text{value}}〉$ and $\mathbf{xfix}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$.

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 $\mathbf{ts}=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{xratio}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xratio}>1.0$.

$\mathbf{ifail}=2$

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

$\mathbf{ifail}=3$

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

In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t=\mathbf{ts}$. The problem may have a singularity, or the error requirement may be inappropriate.

$\mathbf{ifail}=4$

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.