NAG FL Interface
d03prf (dim1_​parab_​remesh_​keller)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs to be solved.

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

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

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

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

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

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

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

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

pdedef must evaluate the functions $G_i$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by d03prf.

The specification of pdedef is:

Fortran Interface copy

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

C Header Interface copy

void  pdedef (const Integer *npde, const double *t, const double *x, const double u[], const double ut[], const double ux[], const Integer *nv, const double v[], const double vdot[], double res[], Integer *ires)

C++ Header Interface copy

#include <nag.h> extern "C" { void  pdedef (const Integer &npde, const double &t, const double &x, const double u[], const double ut[], const double ux[], const Integer &nv, const double v[], const double vdot[], double res[], Integer &ires) }

1: $\mathbf{npde}$ – Integer Input

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) array Input

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

5: $\mathbf{ut}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

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

6: $\mathbf{ux}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

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

7: $\mathbf{nv}$ – Integer Input

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

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

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 $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

9: $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

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 $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

10: $\mathbf{res}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Output

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

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{Q}_{\mathit{i},\mathit{j}}{\stackrel{.}{V}}_{\mathit{j}}\text{,}$$

(9)

i.e., only terms depending explicitly on time derivatives, or

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{Q}_{\mathit{i},\mathit{j}}{\stackrel{.}{V}}_{\mathit{j}}+S_{\mathit{i}}\text{,}$$

(10)

i.e., all terms in equation (3).

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

11: $\mathbf{ires}$ – Integer Input/Output

On entry: the form of $G_i$ that must be returned in the array res.

$\mathbf{ires}=-1$

Equation (9) must be used.

$\mathbf{ires}=1$

Equation (10) must be used.

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 $\mathbf{ires}=3$, d03prf returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03prf 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 d03prf. If your code inadvertently does return any NaNs or infinities, d03prf is likely to produce unexpected results.

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

bndary must evaluate the functions $G_i^L$ and $G_i^R$ which describe the boundary conditions, as given in (5) and (6).

The specification of bndary is:

Fortran Interface copy

Subroutine bndary ( npde, t, ibnd, nobc, u, ut, nv, v, vdot, res, ires) Integer, Intent (In) :: npde, ibnd, nobc, nv Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ut(npde), v(nv), vdot(nv) Real (Kind=nag_wp), Intent (Out) :: res(nobc)

C Header Interface copy

void  bndary (const Integer *npde, const double *t, const Integer *ibnd, const Integer *nobc, const double u[], const double ut[], const Integer *nv, const double v[], const double vdot[], double res[], Integer *ires)

C++ Header Interface copy

#include <nag.h> extern "C" { void  bndary (const Integer &npde, const double &t, const Integer &ibnd, const Integer &nobc, const double u[], const double ut[], const Integer &nv, const double v[], const double vdot[], double res[], Integer &ires) }

1: $\mathbf{npde}$ – Integer Input

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: $\mathbf{ibnd}$ – Integer Input

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

$\mathbf{ibnd}=0$

bndary must compute the left-hand boundary condition at $x=a$.

$\mathbf{ibnd}\ne 0$

bndary must compute of the right-hand boundary condition at $x=b$.

4: $\mathbf{nobc}$ – Integer Input

On entry: specifies the number $n_a$ of boundary conditions at the boundary specified by ibnd.

5: $\mathbf{u}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

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

6: $\mathbf{ut}\left(\mathbf{npde}\right)$ – Real (Kind=nag_wp) array Input

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

7: $\mathbf{nv}$ – Integer Input

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

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

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 $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

9: $\mathbf{vdot}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Input

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: $\mathbf{vdot}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, may only appear linearly as in (11) and (12).

10: $\mathbf{res}\left(\mathbf{nobc}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{res}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $G^L$ or $G^R$, depending on the value of ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{nobc}$, where $G^L$ is defined as

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{H}_{\mathit{i},\mathit{j}}^L{\stackrel{.}{V}}_{\mathit{j}}\text{,}$$

(11)

i.e., only terms depending explicitly on time derivatives, or

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}+\sum _{\mathit{j}=1}^{\mathbf{nv}}{H}_{\mathit{i},\mathit{j}}^L{\stackrel{.}{V}}_{\mathit{j}}+K_{\mathit{i}}^L\text{,}$$

(12)

i.e., all terms in equation (7), and similarly for $G_{\mathit{i}}^R$.

The definitions of $G^L$ and $G^R$ are determined by the input value of ires.

11: $\mathbf{ires}$ – Integer Input/Output

On entry: the form of $G_i^L$ (or $G_i^R$) that must be returned in the array res.

$\mathbf{ires}=-1$

Equation (11) must be used.

$\mathbf{ires}=1$

Equation (12) must be used.

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 $\mathbf{ires}=3$, d03prf returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03prf 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 d03prf. If your code inadvertently does return any NaNs or infinities, d03prf is likely to produce unexpected results.

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

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

The specification of uvinit is:

Fortran Interface copy

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 copy

void  uvinit (const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  uvinit (const Integer &npde, const Integer &npts, const Integer &nxi, const double x[], const double xi[], double u[], const Integer &nv, double v[]) }

1: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

2: $\mathbf{npts}$ – Integer Input

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

3: $\mathbf{nxi}$ – Integer Input

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

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

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) array Input

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

6: $\mathbf{u}(\mathbf{npde},\mathbf{npts})$ – Real (Kind=nag_wp) array Output

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

7: $\mathbf{nv}$ – Integer Input

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

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

On exit: if $\mathbf{nv}>0$, $\mathbf{v}\left(\mathit{i}\right)$ must contain the value of component $V_{\mathit{i}}\left(t_0\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

uvinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03prf 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 d03prf. If your code inadvertently does return any NaNs or infinities, d03prf is likely to produce unexpected results.

7: $\mathbf{u}\left(\mathbf{neqn}\right)$ – Real (Kind=nag_wp) array Input/Output

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

On exit: the computed solution $U_{\mathit{i}}(x_{\mathit{j}},t)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and $V_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,\mathbf{nv}$, evaluated at $t=\mathbf{ts}$, as follows:

• $\mathbf{u}\left(\mathbf{npde}\times (\mathit{j}-1)+\mathit{i}\right)$ contain $U_{\mathit{i}}(x_{\mathit{j}},t)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$ and $\mathit{j}=1,2,\dots ,\mathbf{npts}$, and
• $\mathbf{u}\left(\mathbf{npts}\times \mathbf{npde}+\mathit{i}\right)$ contain $V_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$.

8: $\mathbf{npts}$ – Integer Input

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

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

9: $\mathbf{x}\left(\mathbf{npts}\right)$ – Real (Kind=nag_wp) array Input/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.

10: $\mathbf{nleft}$ – Integer Input

On entry: the number $n_a$ of boundary conditions at the left-hand mesh point $\mathbf{x}\left(1\right)$.

Constraint: $0\le \mathbf{nleft}\le \mathbf{npde}$.

11: $\mathbf{nv}$ – Integer Input

On entry: the number of coupled ODE components.

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

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

odedef must evaluate the functions $R$, which define the system of ODEs, as given in (4).

If you wish to compute the solution of a system of PDEs only (i.e., $\mathbf{nv}=0$), odedef must be the dummy routine d03pek. (d03pek is included in the NAG Library.)

The specification of odedef is:

Fortran Interface copy

Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, ucpt, r, 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), ucpt(npde,nxi) Real (Kind=nag_wp), Intent (Out) :: r(nv)

C Header Interface copy

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 ucpt[], double r[], Integer *ires)

C++ Header Interface copy

#include <nag.h> extern "C" { 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 ucpt[], double r[], Integer &ires) }

1: $\mathbf{npde}$ – Integer Input

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: $\mathbf{nv}$ – Integer Input

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

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

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) array Input

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: $\mathbf{nxi}$ – Integer Input

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

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

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

8: $\mathbf{ucp}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

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

9: $\mathbf{ucpx}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpx}(\mathit{i},\mathit{j})$ contains the value of $\frac{\partial U_{\mathit{i}}(x,t)}{\partial x}$ 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: $\mathbf{ucpt}(\mathbf{npde},\mathbf{nxi})$ – Real (Kind=nag_wp) array Input

On entry: if $\mathbf{nxi}>0$, $\mathbf{ucpt}(\mathit{i},\mathit{j})$ 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 $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

11: $\mathbf{r}\left(\mathbf{nv}\right)$ – Real (Kind=nag_wp) array Output

On exit: if $\mathbf{nv}>0$, $\mathbf{r}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $R$, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$, where $R$ is defined as

$$R=-B\stackrel{.}{V}-C{U}_t^{*}\text{,}$$

(13)

i.e., only terms depending explicitly on time derivatives, or

$$R=A-B\stackrel{.}{V}-C{U}_t^{*}\text{,}$$

(14)

i.e., all terms in equation (4). The definition of $R$ is determined by the input value of ires.

12: $\mathbf{ires}$ – Integer Input/Output

On entry: the form of $R$ that must be returned in the array r.

$\mathbf{ires}=-1$

Equation (13) must be used.

$\mathbf{ires}=1$

Equation (14) 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 $\mathbf{ires}=3$, d03prf returns to the calling subroutine with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

odedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03prf 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 d03prf. If your code inadvertently does return any NaNs or infinities, d03prf is likely to produce unexpected results.

13: $\mathbf{nxi}$ – Integer Input

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) array Input

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

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: $\mathbf{neqn}$ – Integer Input

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) array Input

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) array Input

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.

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

Note: corresponding elements of rtol and atol cannot both be $0.0$.

18: $\mathbf{itol}$ – Integer Input

A value to indicate the form of the local error test. itol indicates to d03prf whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $\Vert e_i/w_i\Vert <1.0$, where $w_i$ is defined as follows:

On entry:

itol	rtol	atol	${\mathit{w}}_{\mathit{i}}$
1	scalar	scalar	$\mathbf{rtol}\left(1\right)\times \left|\mathbf{u}\left(i\right)\right|+\mathbf{atol}\left(1\right)$
2	scalar	vector	$\mathbf{rtol}\left(1\right)\times \left|\mathbf{u}\left(i\right)\right|+\mathbf{atol}\left(i\right)$
3	vector	scalar	$\mathbf{rtol}\left(i\right)\times \left|\mathbf{u}\left(i\right)\right|+\mathbf{atol}\left(1\right)$
4	vector	vector	$\mathbf{rtol}\left(i\right)\times \left|\mathbf{u}\left(i\right)\right|+\mathbf{atol}\left(i\right)$

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: $\mathbf{itol}=1$, $2$, $3$ or $4$.

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

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

while for the maximum norm

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

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) array Input

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 $\mathbf{algopt}\left(1\right)=2.0$, then $\mathbf{algopt}\left(\mathit{i}\right)$, 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, i.e., $\mathbf{laopt}=\text{'S'}$.

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

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 $\mathbf{algopt}\left(29\right)=0.1$.

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

Used as a relative pivot threshold during subsequent Jacobian decompositions (see $\mathbf{algopt}\left(29\right)$) below which an internal error is invoked. $\mathbf{algopt}\left(30\right)$ must be greater than zero, otherwise the default value is used. 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: $\mathbf{remesh}$ – Logical Input

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 d03prf. Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.

23: $\mathbf{nxfix}$ – Integer Input

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) array Input

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.

Constraint: $\mathbf{xfix}\left(\mathit{i}\right)<\mathbf{xfix}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nxfix}-1$, and 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 $\mathbf{x}$ 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: $\mathbf{nrmesh}$ – Integer Input

On entry: indicates the form of meshing to be performed.

$\mathbf{nrmesh}<0$

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 d03prf 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

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

or

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

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 d03prf to force remeshing at several specified times.

28: $\mathbf{ipminf}$ – Integer Input

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

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

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

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

(see Furzeland (1984)). con gives you more control over the mesh distribution e.g., decreasing con allows more clustering. A typical value is $2/(\mathbf{npts}-1)$, 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: $\mathbf{con}=2.0/(\mathbf{npts}-1)$.

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

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 d03pel may be used for monitf. (d03pel is included in the NAG Library.)

The specification of monitf is:

Fortran Interface copy

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

C Header Interface copy

void  monitf (const double *t, const Integer *npts, const Integer *npde, const double x[], const double u[], double fmon[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  monitf (const double &t, const Integer &npts, const Integer &npde, const double x[], const double u[], double fmon[]) }

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

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

2: $\mathbf{npts}$ – Integer Input

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

3: $\mathbf{npde}$ – Integer Input

On entry: the number of PDEs in the system.

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

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: $\mathbf{u}(\mathbf{npde},\mathbf{npts})$ – Real (Kind=nag_wp) array Input

On entry: $\mathbf{u}(\mathit{i},\mathit{j})$ contains the value of $U_{\mathit{i}}(x,t)$ 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: $\mathbf{fmon}\left(\mathbf{npts}\right)$ – Real (Kind=nag_wp) array Output

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

monitf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03prf 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 d03prf. If your code inadvertently does return any NaNs or infinities, d03prf is likely to produce unexpected results.

32: $\mathbf{rsave}\left(\mathbf{lrsave}\right)$ – Real (Kind=nag_wp) array Communication 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: $\mathbf{lrsave}$ – Integer Input

On entry: the dimension of the array rsave as declared in the (sub)program from which d03prf 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 $\mathbf{laopt}=\text{'B'}$, $\mathbf{lrsave}\ge (3\mathit{ml}+\mathit{mu}+2)\times \mathbf{neqn}+\mathit{nwkres}+\mathit{lenode}$.

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

Where $\mathit{ml}$ and $\mathit{mu}$ are the lower and upper half bandwidths given by $\mathit{ml}=\mathbf{npde}+\mathbf{nleft}-1$ such that

for problems involving PDEs only,

$\mathit{mu}=2\mathbf{npde}-\mathbf{nleft}-1\text{;}$

for coupled PDE/ODE problems,

$\mathit{ml}=\mathit{mu}=\mathbf{neqn}-1\text{.}$

Where $\mathit{nwkres}$ is defined by

if $\mathbf{nv}>0\text{ and }\mathbf{nxi}>0$,

$\mathit{nwkres}=\mathbf{npde}(3\mathbf{npde}+6\mathbf{nxi}+\mathbf{npts}+15)+\mathbf{nxi}+\mathbf{nv}+7\mathbf{npts}+\mathbf{nxfix}+1\text{;}$

if $\mathbf{nv}>0\text{ and }\mathbf{nxi}=0$,

$\mathit{nwkres}=\mathbf{npde}(3\mathbf{npde}+\mathbf{npts}+21)+\mathbf{nv}+7\mathbf{npts}+\mathbf{nxfix}+2\text{;}$

if $\mathbf{nv}=0$,

$\mathit{nwkres}=\mathbf{npde}(3\mathbf{npde}+\mathbf{npts}+21)+7\mathbf{npts}+\mathbf{nxfix}+3\text{.}$

Where $\mathit{lenode}$ is defined by

if the BDF method is used,

$\mathit{lenode}=(6+\mathrm{int}\left(\mathbf{algopt}\left(2\right)\right))\times \mathbf{neqn}+50\text{;}$

if the Theta method is used,

$\mathit{lenode}=9\mathbf{neqn}+50\text{.}$

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 array Communication Array

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

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 the following components of the array isave concern the efficiency of the integration:

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

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 evaluating 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: $\mathbf{lisave}$ – Integer Input

On entry: the dimension of the array isave as declared in the (sub)program from which d03prf is called. Its size depends on the type of matrix algebra selected:

• if $\mathbf{laopt}=\text{'F'}$, $\mathbf{lisave}\ge 25+\mathbf{nxfix}$;
• if $\mathbf{laopt}=\text{'B'}$, $\mathbf{lisave}\ge \mathbf{neqn}+25+\mathbf{nxfix}$;
• 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: $\mathbf{itask}$ – Integer Input

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

$\mathbf{itask}=1$

Normal computation of output values u at $t=\mathbf{tout}$ (by overshooting and interpolating).

$\mathbf{itask}=2$

Take one step in the time direction 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: $\mathbf{itrace}$ – Integer Input

On entry: the level of trace information required from d03prf and the underlying ODE solver as follows:

$\mathbf{itrace}\le -1$

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.

$\mathbf{itrace}\ge 3$

The 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: $\mathbf{ind}$ – Integer Input/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 and the remeshing arguments nrmesh, dxmesh, trmesh, xratio and con may be reset between calls to d03prf.

Constraint: $\mathbf{ind}=0$ or $1$.

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

39: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

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

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

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

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, $\mathbf{neqn}=⟨\mathit{\text{value}}⟩$, $\mathbf{npde}=⟨\mathit{\text{value}}⟩$, $\mathbf{npts}=⟨\mathit{\text{value}}⟩$ and $\mathbf{nv}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{neqn}=\mathbf{npde}\times \mathbf{npts}+\mathbf{nv}$.

On entry, $\mathbf{nleft}=⟨\mathit{\text{value}}⟩$, $\mathbf{npde}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nleft}\le \mathbf{npde}$.

On entry, $\mathbf{nleft}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nleft}\ge 0$.

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: $\mathbf{xratio}>1.0$.