NAG Toolbox: nag_pde_1d_parab_convdiff_remesh (d03ps)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs to be solved.

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

2:     $\mathrm{ts}$ – double scalar

The initial value of the independent variable $t$.

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

3:     $\mathrm{tout}$ – double scalar

The final value of $t$ to which the integration is to be carried out.

4:     $\mathrm{pdedef}$ – function handle or string containing name of m-file

pdedef must evaluate the functions $P_{i,j}$, $C_i$, $D_i$ and $S_i$ which partially define the system of PDEs. $P_{i,j}$ and $C_i$ may depend on $x$, $t$, $U$ and $V$; $D_i$ may depend on $x$, $t$, $U$, $U_x$ and $V$; and $S_i$ may depend on $x$, $t$, $U$, $V$ and linearly on $\stackrel{.}{V}$. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_1d_parab_convdiff_remesh (d03ps). The actual argument nag_pde_1d_parab_convdiff_dae_sample_pdedef (d03plp) may be used for pdedef for problems in the form (2). (nag_pde_1d_parab_convdiff_dae_sample_pdedef (d03plp) is included in the NAG Toolbox.)

[p, c, d, s, ires] = pdedef(npde, t, x, u, ux, ncode, v, vdot, ires)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

3:     $\mathrm{x}$ – double scalar

The current value of the space variable $x$.

4:     $\mathrm{u}\left(\mathbf{npde}\right)$ – double array

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

5:     $\mathrm{ux}\left(\mathbf{npde}\right)$ – double array

$\mathbf{ux}\left(\mathit{i}\right)$ 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:     $\mathrm{ncode}$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

7:     $\mathrm{v}\left(\mathbf{ncode}\right)$ – double array

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

8:     $\mathrm{vdot}\left(\mathbf{ncode}\right)$ – double array

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

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

9:     $\mathrm{ires}$ – int64int32nag_int scalar

Set to $-1\text{ or }1$.

Output Parameters

1:     $\mathrm{p}\left(\mathbf{npde},\mathbf{npde}\right)$ – double array

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

2:     $\mathrm{c}\left(\mathbf{npde}\right)$ – double array

$\mathbf{c}\left(\mathit{i}\right)$ must be set to the value of $C_{\mathit{i}}\left(x,t,U,V\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

3:     $\mathrm{d}\left(\mathbf{npde}\right)$ – double array

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

4:     $\mathrm{s}\left(\mathbf{npde}\right)$ – double array

$\mathbf{s}\left(\mathit{i}\right)$ must be set to the value of $S_{\mathit{i}}\left(x,t,U,V,\stackrel{.}{V}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:     $\mathrm{ires}$ – int64int32nag_int scalar

Should usually remain unchanged. However, you may set ires to force the integration function 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 $\mathbf{ires}=3$, then nag_pde_1d_parab_convdiff_remesh (d03ps) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

5:     $\mathrm{numflx}$ – function handle or string containing name of m-file

numflx must supply the numerical flux for each PDE given the left and right values of the solution vector $\mathbf{u}$. numflx is called approximately midway between each pair of mesh points in turn by nag_pde_1d_parab_convdiff_remesh (d03ps).

[flux, ires] = numflx(npde, t, x, ncode, v, uleft, uright, ires)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

3:     $\mathrm{x}$ – double scalar

The current value of the space variable $x$.

4:     $\mathrm{ncode}$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

5:     $\mathrm{v}\left(\mathbf{ncode}\right)$ – double array

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

6:     $\mathrm{uleft}\left(\mathbf{npde}\right)$ – double array

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

7:     $\mathrm{uright}\left(\mathbf{npde}\right)$ – double array

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

8:     $\mathrm{ires}$ – int64int32nag_int scalar

Set to $-1\text{ or }1$.

Output Parameters

1:     $\mathrm{flux}\left(\mathbf{npde}\right)$ – double array

$\mathbf{flux}\left(\mathit{i}\right)$ must be set to the numerical flux ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

2:     $\mathrm{ires}$ – int64int32nag_int scalar

Should usually remain unchanged. However, you may set ires to force the integration function 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 $\mathbf{ires}=3$, then nag_pde_1d_parab_convdiff_remesh (d03ps) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

6:     $\mathrm{bndary}$ – function handle or string containing name of m-file

bndary must evaluate the functions $G_i^L$ and $G_i^R$ which describe the physical and numerical boundary conditions, as given by (8) and (9).

[g, ires] = bndary(npde, npts, t, x, u, ncode, v, vdot, ibnd, ires)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{npts}$ – int64int32nag_int scalar

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

3:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

4:     $\mathrm{x}\left(\mathbf{npts}\right)$ – double array

The mesh points in the spatial direction. $\mathbf{x}\left(1\right)$ corresponds to the left-hand boundary, $a$, and $\mathbf{x}\left(\mathbf{npts}\right)$ corresponds to the right-hand boundary, $b$.

5:     $\mathrm{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – double array

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

Note:  if banded matrix algebra is to be used then the functions $G_{\mathit{i}}^L$ and $G_{\mathit{i}}^R$ may depend on the value of $U_{\mathit{i}}\left(x,t\right)$ at the boundary point and the two adjacent points only.

6:     $\mathrm{ncode}$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

7:     $\mathrm{v}\left(\mathbf{ncode}\right)$ – double array

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

8:     $\mathrm{vdot}\left(\mathbf{ncode}\right)$ – double array

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

Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{ncode}$, may only appear linearly in $G_{\mathit{j}}^L$ and $G_{\mathit{j}}^R$, for $\mathit{j}=1,2,\dots ,\mathbf{npde}$.

9:     $\mathrm{ibnd}$ – int64int32nag_int scalar

Specifies which boundary conditions are to be evaluated.

$\mathbf{ibnd}=0$

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

$\mathbf{ibnd}\ne 0$

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

10:   $\mathrm{ires}$ – int64int32nag_int scalar

Set to $-1\text{ or }1$.

Output Parameters

1:     $\mathrm{g}\left(\mathbf{npde}\right)$ – double array

$\mathbf{g}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of either $G_{\mathit{i}}^L$ or $G_{\mathit{i}}^R$ in (8) and (9), depending on the value of ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

2:     $\mathrm{ires}$ – int64int32nag_int scalar

Should usually remain unchanged. However, you may set ires to force the integration function 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 $\mathbf{ires}=3$, then nag_pde_1d_parab_convdiff_remesh (d03ps) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

7:     $\mathrm{uvinit}$ – function handle or string containing name of m-file

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 $a\le x\le b$.

[u, v] = uvinit(npde, npts, nxi, x, xi, ncode)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{npts}$ – int64int32nag_int scalar

The number of mesh points in the interval [$a,b$].

3:     $\mathrm{nxi}$ – int64int32nag_int scalar

The number of ODE/PDE coupling points.

4:     $\mathrm{x}\left(\mathbf{npts}\right)$ – double array

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:     $\mathrm{xi}\left(\mathbf{nxi}\right)$ – double array

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:     $\mathrm{ncode}$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

Output Parameters

1:     $\mathrm{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – double array

If $\mathbf{nxi}>0$, $\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ 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 $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

2:     $\mathrm{v}\left(\mathbf{ncode}\right)$ – double array

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

8:     $\mathrm{u}\left(\mathbf{neqn}\right)$ – double array

If $\mathbf{ind}=1$ the value of u must be unchanged from the previous call.

9:     $\mathrm{x}\left(\mathbf{npts}\right)$ – double array

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

10:   $\mathrm{ncode}$ – int64int32nag_int scalar

The number of coupled ODE components.

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

11:   $\mathrm{odedef}$ – function handle or string containing name of m-file

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., $\mathbf{ncode}=0$), odedef must be the string nag_pde_1d_parab_dae_keller_remesh_fd_dummy_odedef (d03pek). (nag_pde_1d_parab_dae_keller_remesh_fd_dummy_odedef (d03pek) is included in the NAG Toolbox.)

[r, ires] = odedef(npde, t, ncode, v, vdot, nxi, xi, ucp, ucpx, ucpt, ires)

Input Parameters

1:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

2:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

3:     $\mathrm{ncode}$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

4:     $\mathrm{v}\left(\mathbf{ncode}\right)$ – double array

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

5:     $\mathrm{vdot}\left(\mathbf{ncode}\right)$ – double array

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

6:     $\mathrm{nxi}$ – int64int32nag_int scalar

The number of ODE/PDE coupling points.

7:     $\mathrm{xi}\left(\mathbf{nxi}\right)$ – double array

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:     $\mathrm{ucp}\left(\mathbf{npde},:\right)$ – double array

The second dimension of the array ucp must be at least $\max \left(1,\mathbf{nxi}\right)$.

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:     $\mathrm{ucpx}\left(\mathbf{npde},:\right)$ – double array

The second dimension of the array ucpx must be at least $\max \left(1,\mathbf{nxi}\right)$.

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:   $\mathrm{ucpt}\left(\mathbf{npde},:\right)$ – double array

The second dimension of the array ucpt must be at least $\max \left(1,\mathbf{nxi}\right)$.

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 $\mathit{j}=1,2,\dots ,\mathbf{nxi}$.

11:   $\mathrm{ires}$ – int64int32nag_int scalar

The form of $R$ that must be returned in the array r.

$\mathbf{ires}=1$

Equation (10) must be used.

$\mathbf{ires}=-1$

Equation (11) must be used.

Output Parameters

1:     $\mathrm{r}\left(\mathbf{ncode}\right)$ – double array

$\mathbf{r}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $R$, for $\mathit{i}=1,2,\dots ,\mathbf{ncode}$, where $R$ is defined as

$$R=L-M\stackrel{.}{V}-N{U}_t^{*}\text{,}$$

(10)

or

$$R=-M\stackrel{.}{V}-N{U}_t^{*}\text{.}$$

(11)

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

2:     $\mathrm{ires}$ – int64int32nag_int scalar

Should usually remain unchanged. However, you may reset ires to force the integration function 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 $\mathbf{ires}=3$, then nag_pde_1d_parab_convdiff_remesh (d03ps) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

12:   $\mathrm{xi}\left(\mathbf{nxi}\right)$ – double array

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

13:   $\mathrm{rtol}\left(:\right)$ – double array

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 $4$

The relative local error tolerance.

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

14:   $\mathrm{atol}\left(:\right)$ – double array

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 $4$

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 $0.0$.

15:   $\mathrm{itol}$ – int64int32nag_int scalar

A value to indicate the form of the local error test. If $e_{\mathit{i}}$ is the estimated local error for $\mathbf{u}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{neqn}$, and $‖‖$, denotes the norm, then the error test to be satisfied is $‖e_{\mathit{i}}‖<1.0$. itol indicates to nag_pde_1d_parab_convdiff_remesh (d03ps) whether to interpret either or both of rtol and atol as a vector or scalar in the formation of the weights $w_{\mathit{i}}$ used in the calculation of the norm (see the description of norm_p):

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

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

16:   $\mathrm{norm\_p}$ – string (length ≥ 1)

The type of norm to be used.

$\mathbf{norm\_p}=\text{'1'}$

Averaged $L_1$ norm.

$\mathbf{norm\_p}=\text{'2'}$

Averaged $L_2$ norm.

If $U_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged $L_1$ norm

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

and for the averaged $L_2$ norm

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

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

Constraint: $\mathbf{norm\_p}=\text{'1'}$ or $\text{'2'}$.

17:   $\mathrm{laopt}$ – string (length ≥ 1)

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 ($\mathbf{ncode}=0$). Also, the banded option should not be used if the boundary conditions involve solution components at points other than the boundary and the immediately adjacent two points.

18:   $\mathrm{algopt}\left(30\right)$ – double array

May be set to control various options available in the integrator. If you wish to employ all the default options, then $\mathbf{algopt}\left(1\right)$ 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 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, then 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 $\mathbf{algopt}\left(4\right)=1.0$, then the Petzold test is used. If $\mathbf{algopt}\left(4\right)=2.0$, then the Petzold test is not used. The default value is $\mathbf{algopt}\left(4\right)=1.0$.

If $\mathbf{algopt}\left(1\right)=1.0$, then $\mathbf{algopt}\left(\mathit{i}\right)$, 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 $\mathbf{algopt}\left(7\right)=1.0$, then switching is allowed and if $\mathbf{algopt}\left(7\right)=2.0$, then 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 $\mathbf{algopt}\left(14\right)=0.0$, then 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 $\mathbf{algopt}\left(15\right)=0.0$, then 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 the range then the default value is used. If the functions 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 the 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 matrix is found to be numerically singular (see $\mathbf{algopt}\left(29\right)$). The default value is $\mathbf{algopt}\left(30\right)=0.0001$.

19:   $\mathrm{remesh}$ – logical scalar

Indicates whether or not spatial remeshing should be performed.

$\mathbf{remesh}=\mathit{true}$

Indicates that spatial remeshing should be performed as specified.

$\mathbf{remesh}=\mathit{false}$

Indicates that spatial remeshing should be suppressed.

Note:  remesh should not be changed between consecutive calls to nag_pde_1d_parab_convdiff_remesh (d03ps). Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.

20:   $\mathrm{xfix}\left(:\right)$ – double array

The dimension of the array xfix must be at least $\max \left(1,\mathbf{nxfix}\right)$

$\mathbf{xfix}\left(\mathit{i}\right)$, 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 $\mathit{j}$, $2\le \mathit{j}\le \mathbf{npts}-1$..

Note: the positions of the fixed mesh points in the array $\mathbf{x}\left(\mathbf{npts}\right)$ 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.

21:   $\mathrm{nrmesh}$ – int64int32nag_int scalar

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 nag_pde_1d_parab_convdiff_remesh (d03ps) to give greater flexibility over the times of remeshing.

22:   $\mathrm{dxmesh}$ – double scalar

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

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

or

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

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

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

23:   $\mathrm{trmesh}$ – double scalar

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 nag_pde_1d_parab_convdiff_remesh (d03ps) to force remeshing at several specified times.

24:   $\mathrm{ipminf}$ – int64int32nag_int scalar

The level of trace information regarding the adaptive remeshing. Details are directed to the current advisory message unit (see nag_file_set_unit_advisory (x04ab)).

$\mathbf{ipminf}=0$

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 $2$.

25:   $\mathrm{monitf}$ – function handle or string containing name of m-file

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

If you specify $\mathbf{remesh}=\mathit{false}$, i.e., no remeshing, then monitf will not be called and the string nag_pde_1d_parab_dae_keller_remesh_fd_dummy_monitf (d03pel) may be used for monitf. (nag_pde_1d_parab_dae_keller_remesh_fd_dummy_monitf (d03pel) is included in the NAG Toolbox.)

[fmon] = monitf(t, npts, npde, x, u)

Input Parameters

1:     $\mathrm{t}$ – double scalar

The current value of the independent variable $t$.

2:     $\mathrm{npts}$ – int64int32nag_int scalar

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

3:     $\mathrm{npde}$ – int64int32nag_int scalar

The number of PDEs in the system.

4:     $\mathrm{x}\left(\mathbf{npts}\right)$ – double array

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:     $\mathrm{u}\left(\mathbf{npde},\mathbf{npts}\right)$ – double array

$\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ 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 $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

Output Parameters

1:     $\mathrm{fmon}\left(\mathbf{npts}\right)$ – double array

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

26:   $\mathrm{rsave}\left(\mathit{lrsave}\right)$ – double 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 function because it contains required information about the iteration.

27:   $\mathrm{isave}\left(\mathit{lisave}\right)$ – int64int32nag_int array

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

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the function 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 BDF method last used in the time integration, if applicable. When the Theta method is used, $\mathbf{isave}\left(4\right)$ contains no useful information.

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

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.

28:   $\mathrm{itask}$ – int64int32nag_int scalar

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 $5$.

29:   $\mathrm{itrace}$ – int64int32nag_int scalar

The level of trace information required from nag_pde_1d_parab_convdiff_remesh (d03ps) and the underlying ODE solver. itrace may take the value $-1$, $0$, $1$, $2$ or $3$.

$\mathbf{itrace}=-1$

No output is generated.

$\mathbf{itrace}=0$

Only warning messages from the PDE solver are printed on the current error message unit (see nag_file_set_unit_error (x04aa)).

$\mathbf{itrace}>0$

Output from the underlying ODE solver is printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

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

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

30:   $\mathrm{ind}$ – int64int32nag_int scalar

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 function. In this case, only the arguments tout, ifail, nrmesh and trmesh may be reset between calls to nag_pde_1d_parab_convdiff_remesh (d03ps).

Constraint: $\mathbf{ind}=0$ or $1$.

Optional Input Parameters

1:     $\mathrm{npts}$ – int64int32nag_int scalar

Default: the dimension of the array x.

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

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

2:     $\mathrm{nxi}$ – int64int32nag_int scalar

Default: the dimension of the array xi.

The number of ODE/PDE coupling points.

Constraints:

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

3:     $\mathrm{neqn}$ – int64int32nag_int scalar

Default: the dimension of the array u.

The number of ODEs in the time direction.

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

4:     $\mathrm{nxfix}$ – int64int32nag_int scalar

Default: the dimension of the array xfix.

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.

5:     $\mathrm{xratio}$ – double scalar

Default: $1.5$

An input bound on the adjacent mesh ratio (greater than $1.0$ and typically in the range $1.5$ to $3.0$). The remeshing functions will attempt to ensure that

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

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

6:     $\mathrm{con}$ – double scalar

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

Default: $2.0/\left(\mathbf{npts}-1\right)$

An input bound on the sub-integral of the monitor function $F^{\mathrm{mon}}\left(x\right)$ over each space step. The remeshing functions 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 $2.0/\left(\mathbf{npts}-1\right)$, 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.

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

Output Parameters

1:     $\mathrm{ts}$ – double scalar

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

2:     $\mathrm{u}\left(\mathbf{neqn}\right)$ – double array

$\mathbf{u}\left(\mathbf{npde}\times \left(\mathit{j}-1\right)+\mathit{i}\right)$ 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 $\mathit{k}=1,2,\dots ,\mathbf{ncode}$, all evaluated at $t=\mathbf{ts}$.

3:     $\mathrm{x}\left(\mathbf{npts}\right)$ – double array

The final values of the mesh points.

4:     $\mathrm{rsave}\left(\mathit{lrsave}\right)$ – double array

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

5:     $\mathrm{isave}\left(\mathit{lisave}\right)$ – int64int32nag_int array

If $\mathbf{ind}=1$, isave must be unchanged from the previous call to the function 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 BDF method last used in the time integration, if applicable. When the Theta method is used, $\mathbf{isave}\left(4\right)$ contains no useful information.

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

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.

6:     $\mathrm{ind}$ – int64int32nag_int scalar

$\mathbf{ind}=1$.

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

On entry,	$\mathbf{ts}\ge \mathbf{tout}$,
or	$\mathbf{tout}-\mathbf{ts}$ is too small,
or	$\mathbf{itask}\ne 1$, $2$, $3$, $4$ or $5$,
or	at least one of the coupling points defined in array xi is outside the interval [$\mathbf{x}\left(1\right),\mathbf{x}\left(\mathbf{npts}\right)$],
or	the coupling points are not in strictly increasing order,
or	$\mathbf{npts}<3$,
or	$\mathbf{npde}<1$,
or	$\mathbf{laopt}\ne \text{'F'}$, $\text{'B'}$ or $\text{'S'}$,
or	$\mathbf{itol}\ne 1$, $2$, $3$ or $4$,
or	$\mathbf{ind}\ne 0$ or $1$,
or	mesh points $\mathbf{x}\left(i\right)$ are badly ordered,
or	lrsave is too small,
or	lisave is too small,
or	ncode and nxi are incorrectly defined,
or	$\mathbf{ind}=1$ on initial entry to nag_pde_1d_parab_convdiff_remesh (d03ps),
or	$\mathbf{neqn}\ne \mathbf{npde}\times \mathbf{npts}+\mathbf{ncode}$,
or	an element of rtol or $\mathbf{atol}<0.0$,
or	corresponding elements of rtol and atol are both $0.0$,
or	$\mathbf{norm\_p}\ne \text{'1'}$ or $\text{'2'}$,
or	nxfix not in the range $0$ to $\mathbf{npts}-2$,
or	fixed mesh point(s) do not coincide with any of the user-supplied mesh points,
or	$\mathbf{dxmesh}<0.0$,
or	$\mathbf{ipminf}\ne 0$, $1$ or $2$,
or	$\mathbf{xratio}\le 1.0$,
or	con not in the range $0.1/\left(\mathbf{npts}-1\right)$ to $10.0/\left(\mathbf{npts}-1\right)$.

W  $\mathbf{ifail}=2$

The underlying ODE solver cannot make any further progress, with the values of atol and rtol, across the integration range from the current point $t=\mathbf{ts}$. The components of u contain the computed values at the current point $t=\mathbf{ts}$.

W  $\mathbf{ifail}=3$

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. Incorrect specification of boundary conditions may also result in this error.

   $\mathbf{ifail}=4$

In setting up the ODE system, the internal initialization function was unable to initialize the derivative of the ODE system. This could be due to the fact that ires was repeatedly set to $3$ in one of pdedef, numflx, bndary or odedef, when the residual in the underlying ODE solver was being evaluated. Incorrect specification of boundary conditions may also result in this error.

   $\mathbf{ifail}=5$

In solving the ODE system, a singular Jacobian has been encountered. Check the problem formulation.

W  $\mathbf{ifail}=6$

When evaluating the residual in solving the ODE system, ires was set to $2$ in at least one of pdedef, numflx, bndary or odedef. Integration was successful as far as $t=\mathbf{ts}$.

   $\mathbf{ifail}=7$

The values of atol and rtol are so small that the function is unable to start the integration in time.

   $\mathbf{ifail}=8$

In one of pdedef, numflx, bndary or odedef, ires was set to an invalid value.

   $\mathbf{ifail}=9$ (nag_ode_ivp_stiff_imp_revcom (d02nn))

A serious error has occurred in an internal call to the specified function. Check the problem specification and all arguments and array dimensions. Setting $\mathbf{itrace}=1$ may provide more information. If the problem persists, contact NAG.

W  $\mathbf{ifail}=10$

The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $\mathbf{itask}\ne 2$ or $5$.)

   $\mathbf{ifail}=11$

An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current advisory message unit when $\mathbf{itrace}\ge 1$). If using the sparse matrix algebra option, the values of $\mathbf{algopt}\left(29\right)$ and $\mathbf{algopt}\left(30\right)$ may be inappropriate.

   $\mathbf{ifail}=12$

In solving the ODE system, the maximum number of steps specified in $\mathbf{algopt}\left(15\right)$ have been taken.

W  $\mathbf{ifail}=13$

Some error weights $w_i$ became zero during the time integration (see the description of itol). Pure relative error control ($\mathbf{atol}\left(i\right)=0.0$) was requested on a variable (the $i$th) which has become zero. The integration was successful as far as $t=\mathbf{ts}$.

   $\mathbf{ifail}=14$

One or more of the functions $P_{i,j}$, $D_i$ or $C_i$ was detected as depending on time derivatives, which is not permissible.

   $\mathbf{ifail}=15$

When using the sparse option, the value of lisave or lrsave was not sufficient (more detailed information may be directed to the current error message unit, see nag_file_set_unit_error (x04aa)).

   $\mathbf{ifail}=16$

remesh has been changed between calls to nag_pde_1d_parab_convdiff_remesh (d03ps).

   $\mathbf{ifail}=17$

fmon is negative at one or more mesh points, or zero mesh spacing has been obtained due to an inappropriate choice of monitor function.

   $\mathbf{ifail}=-99$

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

   $\mathbf{ifail}=-399$

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

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d03ps_example

function d03ps_example


fprintf('d03ps example results\n\n');


global usav1 usav2;
fprintf(' Example 1\n\n');
ex1;
% Plot results.
fig1 = figure;
ex1_plot;
fprintf('\n\n\n Example 2\n\n');
ex2;
% Plot results.
fig2 = figure;
ex2_plot;


function ex1
  % First d03ps example.  Advection and diffusion of a cloud of material.
  global xsav1 tsav1 usav1;

  % Set values for problem parameters.
  ncode  = int64(0);
  nxi    = int64(0);
  npde   = int64(1);

  % Number of points on calculation mesh, and on interpolated mesh.
  npts   = 61;

  neqn   = npde*npts + ncode;
  lisave = 25 + neqn;
  lrsave = 5000;

  % Define some arrays.
  algopt = zeros(30, 1);
  u      = zeros(npde, npts);
  x      = [0:1/(npts-1):1];
  rsave  = zeros(lrsave, 1);
  isave  = zeros(lisave, 1, 'int64');

  itrace = int64(0);
  itol   = int64(1);
  norm_p = '1';
  atol   = [0.0001];
  rtol   = [0.0001];

  xfix   = [];

  remesh = true;
  nrmesh = int64(3);
  dxmesh = 0.0;
  trmesh = 0.0;
  con    = 2.0/(npts-1.0);
  xratio = 1.5;
  ipminf = int64(0);
  xi     = [0.0];
  laopt  = 'B';

  % b.d.f. integration.
  algopt = zeros(30,1);
  algopt(1)  = 1.0;
  algopt(13) = 0.005;

  % We run through the calculation twice - once to collect the results
  % for plotting, and once to do interpolation to a coarser grid and
  % and output the results.  Set ind = 0 to (re)start the integration
  % in time, and set time to its initial value. Set itask = 2, so that
  % the integrator takes one time-step at a time.
  ind   = int64(0);
  ts    = 0.0;
  itask = int64(2);
  tout  = 0.3;
  tnext = 0.0;

  fprintf(' npts = %4d atol = %10.3e  rtol = %10.3e \n\n', npts, atol, rtol);
  isav  = 0;
  while ts < tout
    [ts, u, x, rsave, isave, ind, ifail] = ...
    d03ps( ...
           npde, ts, tout, @ex1_pdedef, @ex1_numflx, @ex1_bndary, ...
           @ex1_uvinit, u, x, ncode, 'd03pek', xi, rtol, atol, itol, ...
           norm_p, laopt, algopt, remesh, xfix, nrmesh, dxmesh, ...
           trmesh, ipminf, @ex1_monitf, rsave, isave, itask, itrace, ...
           ind, 'nxi', nxi, 'xratio', xratio, 'con', con);
    
    if ts >= tnext
      % Save this timestep, and this set of results.  Note we have to
      % save the x coordinates at this timestep as well, because the
      % algorithm is automatically remeshing the spatial grid.
      isav = isav+1;
      tsav1(isav) = ts;
      for i=1:npts
        usav1(isav,i) = u(1,i);
        xsav1(isav,i) = x(i);
      end
      tnext = tnext + 0.01;
    end
  end

  % Now prepare to run through the calculation again.  First, set up
  % the points on the interpolation grid.
  intpts = 7;
  for i = 0:intpts - 1
      xinterp(i+1) = 0.2 + i*0.1;
  end

  % Set ind = 0 to (re)start the integration in time, set the time to its
  % initial value, and set itask = 1 to compute solution at t=tout by
  % overshooting & interpolating. Reset the calculation mesh, which had been
  % modified by the earlier calls to d03ps.
  ind   = int64(0);
  itask = int64(1);
  ts = 0.0;
  dx = 1/(npts-1);
  x = [0:dx:1];

  for it = 1:3
    tout = 0.1*it;
    [ts, u, x, rsave, isave, ind, ifail] = ...
    d03ps( ...
           npde, ts, tout, @ex1_pdedef, @ex1_numflx, @ex1_bndary, ...
           @ex1_uvinit, u, x, ncode, 'd03pek', xi, rtol, atol, itol, ...
           norm_p, laopt, algopt, remesh, xfix, nrmesh, dxmesh, trmesh, ...
           ipminf, @ex1_monitf, rsave, isave, itask, itrace, ...
           ind, 'nxi', nxi, 'xratio', xratio, 'con', con);

    fprintf(' t = %6.3f\n', ts);
    fprintf(' x       ');
    fprintf('%9.4f', xinterp);
    fprintf('\n');

    % Call d03pz to do interpolation of results onto coarser grid.
    m = int64(0);
    itype = int64(1);
    [uinterp, ifail] = d03pz( ...
                              m, u, x, xinterp, itype);
    fprintf(' approx u');
    fprintf('%9.4f', uinterp);
    fprintf('\n\n');
  end

  % Output some statistics.
  fprintf([' Number of integration steps in time = %6d\n', ...
           ' Number of function evaluations = %6d\n', ...
           ' Number of Jacobian evaluations = %6d\n', ...
           ' Number of iterations = %6d\n'], ...
          isave(1), isave(2), isave(3), isave(5));

function [p, c, d, s, ires] = ex1_pdedef(npde, t, x, u, ux, ncode, v, ...
                                         vdot, ires)
  p = zeros(npde, npde);
  c = zeros(npde, 1);
  d = zeros(npde, 1);
  s = zeros(npde, 1);

  p(1,1) = 1;
  c(1) = 0.2d-2;
  d(1) = ux(1);


function [flux, ires] = ex1_numflx(npde, t, x, ncode, v, uleft, uright, ires)
  flux = zeros(npde, 1);
  flux(1) = uleft(1);


function [g, ires] = ex1_bndary(npde, npts, t, x, u, ncode, v, vdot, ibnd, ires)
  g = zeros(npde, 1);

  if (ibnd == 0)
    g(1) = u(1,1);
  else
    g(1) = u(1,npts);
  end


function [u, v] = ex1_uvinit(npde, npts, nxi, x, xi, ncode)
  u = zeros(npde, npts);
  v = zeros(ncode, 1);

  for i = 1:double(npts)
    if (x(i) > 0.2 && x(i) <= 0.4)
      u(1,i) = sin(pi*(5*x(i)-1));
    end
  end


function [r, ires] = ex1_odedef(npde, t, ncode, v, vdot, nxi, xi, ucp, ucpx, ...
                                ucpt, ires)
  r = zeros(ncode, 1);


function [fmon] = ex1_monitf(t, npts, npde, x, u)
  fmon = zeros(npts, 1);

  for i = 2:double(npts) - 1
    h1 = x(i) - x(i-1);
    h2 = x(i+1) - x(i);
    h3 = (x(i+1)-x(i-1))/2;
    % second derivatives ..
    fmon(i) = abs(((u(1,i+1)-u(1,i))/h2-(u(1,i)-u(1,i-1))/h1)/h3);
  end
  fmon(1) = fmon(2);
  fmon(npts) = fmon(double(npts)-1);


function ex2
  % Second d03ps example. Linear advection equation w/ non-linear source term.
  global xsav2 tsav2 usav2;

  % Set values for problem parameters.
  ncode = int64(0);
  nxi   = int64(0);
  npde  = int64(1);

  % Number of points on calculation mesh, and on interpolated mesh.
  npts = 61;

  neqn   = npde*npts + ncode;
  lisave = 25 + neqn;
  lrsave = 5000;

  % Define some arrays.
  algopt = zeros(30, 1);
  u = zeros(npde, npts);
  x = zeros(npts, 1);
  rsave = zeros(lrsave, 1);
  isave = zeros(lisave, 1, 'int64');

  itrace = int64(0);
  itol = int64(1);
  norm_p = '1';
  atol = [0.0005];
  rtol = [0.05];

  % Initialize calculation mesh.
  dx = 1/(npts-1);
  x = [0:dx:1];
  xfix = [];

  % Set remesh parameters.
  remesh = true;
  nrmesh = int64(5);
  dxmesh = 0;
  trmesh = 0;
  con = dx;
  xratio = 1.5;
  ipminf = int64(0);
  xi = [0.0];
  laopt = 'B';

  % b.d.f. integration.
  algopt = zeros(30,1);
  % Theta integration.
  algopt(1) = 2.0;
  algopt(6) = 2.0;
  algopt(7) = 2.0;
  % Maximum time step.
  algopt(13) = 0.0025;

  % We run through the calculation twice - once to collect the results for
  % plotting, and once to do interpolation to a coarser grid and and output
  % the results.  Set ind = 0 to (re)start the integration in time, and set
  % the time to its initial value. Set itask = 2, tp take one time-step at
  % a time.
  ind = int64(0);
  ts = 0.0;
  itask = int64(2);
  tout = 0.4;
  tnext = 0.0;

  isav = 0;
  while ts < tout
    [ts, u, x, rsave, isave, ind, ifail] = ...
    d03ps( ...
           npde, ts, tout, @ex2_pdedef, @ex2_numflx, @ex2_bndary, ...
           @ex2_uvinit, u, x, ncode, 'd03pek', xi, rtol, atol, itol, ...
           norm_p, laopt, algopt, remesh, xfix, nrmesh, dxmesh, trmesh, ...
           ipminf, @ex2_monitf, rsave, isave, itask, itrace, ...
           ind, 'nxi', nxi, 'xratio', xratio, 'con', con);
    
    if ts >= tnext
      % Save this timestep, and this set of results.  Note we have to
      % save the x coordinates at this timestep as well, because the
      % algorithm is automatically remeshing the spatial grid.
      isav = isav+1;
      tsav2(isav) = ts;
      usav2(isav,1:npts) = u(1,:);
      xsav2(isav,1:npts) = x;
      tnext = tnext + 0.01;
    end
  end

  % Now prepare to run through the calculation again.  First, set up
  % the points on the interpolation grid.
  intpts = 7;
  xinterp = [0.2:0.1:0.8];

  % We set ind = 0 to (re)start the integration in time, set the time to its
  % initial value, and set itask = 1 to compte solution at t=tout by
  % overshooting & interpolating.
  % We also need to reset the calculation mesh, since this has been modified
  % by the earlier calls to d03ps.
  ind   = int64(0);
  itask = int64(1);
  ts = 0;
  dx = 1/(npts-1);
  x = [0:dx:1];

  for it = 1:2
    tout = 0.2*it;
    [ts, u, x, rsave, isave, ind, ifail] = ...
    d03ps( ...
           npde, ts, tout, @ex2_pdedef, @ex2_numflx, @ex2_bndary, ...
           @ex2_uvinit, u, x, ncode, 'd03pek', xi, rtol, atol, itol, ...
           norm_p, laopt, algopt, remesh, xfix, nrmesh, dxmesh, trmesh, ...
           ipminf, @ex2_monitf, rsave, isave, itask, itrace, ...
           ind, 'nxi', nxi, 'xratio', xratio, 'con', con);

    if it == 1
      fprintf(' npts = %4d atol = %10.3e  rtol = %10.3e \n\n', npts, ...
              atol, rtol)
    end
    fprintf(' t = %6.3f\n', ts);

    % Call d03pz to do interpolation of results onto coarser grid.
    m = int64(0);
    itype = int64(1);
    [uinterp, ifail] = d03pz(m, u, x, xinterp, itype);

    % Check against exact solution.
    [ue] = ex2_exact(tout, xinterp, intpts);
    fprintf('        x      approx u    exact u\n');
    for i=1:intpts
      fprintf('%9.4f   %9.4f   %9.4f\n', xinterp(i), uinterp(i), ue(i));
    end
    fprintf('\n\n');
  end

  % Output some statistics.
  fprintf([' Number of integration steps in time = %6d\n', ...
           ' Number of function evaluations = %6d\n', ...
           ' Number of Jacobian evaluations = %6d\n', ...
           ' Number of iterations = %6d\n'], ...
          isave(1), isave(2), isave(3), isave(5));


function [p, c, d, s, ires] = ex2_pdedef(npde, t, x, u, ux, ...
      ncode, v, vdot, ires)
  % Evaluate pij, ci, di, si to partially define the system of PDEs.

  p = zeros(npde, npde);
  c = zeros(npde, 1);
  d = zeros(npde, 1);
  s = zeros(npde, 1);

  p(1,1) = 1;
  c(1) = 0;
  d(1) = 0;
  s(1) = -100*u(1)*(u(1) - 1)*(u(1) - 0.5);

function [fmon] = ex2_monitf(t, npts, npde, x, u)
  % Supplies and evaluates a remesh monitor function.
  persistent icount;
  persistent xa;

  fmon = zeros(npts, 1);

  % Locate shock.
  uxmax = 0;
  xmax = 0;
  for i = 2:npts-1
    h1 = x(i) - x(i-1);
    ux = abs((u(1,i) - u(1,i-1))/h1);
    if (ux > uxmax)
      uxmax = ux;
      xmax = x(i);
    end
  end

  % Assign width (on first call only).
  if (isempty(icount))
    icount = 1;
    xleft = xmax - x(1);
    xright = x(npts) - xmax;
    if (xleft > xright)
      xa = xright;
    else
      xa = xleft;
    end
  end

  % Assign monitor function.
  xl = xmax - xa;
  xr = xmax + xa;
  for i = 1:npts
    if ((x(i) > xl) && (x(i) < xr))
      fmon(i) = 1.0 + cos(pi*(x(i) - xmax)/xa);
    else
      fmon(i) = 0.0;
    end
  end

function [g, ires] = ex2_bndary(npde, npts, t, x, u, ncode, v, ...
                                vdot, ibnd, ires)
  % Evaluate giL and giR to describe the physical & numerical boundary
  % conditions.

  % Solution known to be constant at both boundaries.
  if (ibnd == 0)
    [ue] = ex2_exact(t, x(1), 1);
    g(1) = ue(1,1) - u(1,1);
  else
    [ue] = ex2_exact(t, x(npts), 1);
    g(1) = ue(1,1) - u(1,npts);
  end

function [flux, ires] = ex2_numflx(npde, t, x, ncode, v, uleft, ...
                                   uright, ires)
  % Supplies numerical flux for each PDE given the L & R values of u.
  flux = zeros(npde, 1);
  flux(1) = uleft(1);


function [u, v] = ex2_uvinit(npde, npts, nxi, x, xi, ncode)
  % Specify initial values (at t=t0) of u(x,t) and v(t) for all x.

  u = zeros(npde, npts);
  v = zeros(ncode, 1);
  t = 0.0;
  [u] = ex2_exact(t, x, npts);

function [u] = ex2_exact(t, x, npts)
  % Calculate exact solution (for comparison and use in boundary conditions).

  s = 0.1;
  del = 0.01;
  rm = -1.0/del;
  rn = 1.0 + s/del;

  for i = 1:npts
    psi = x(i) - t;
    if (psi < s)
      u(1,i) = 1.0;
    elseif (psi > (del+s))
      u(1,i) = 0.0;
    else
      u(1,i) = rm*psi + rn;
    end
  end

function ex1_plot
  % Plot the results.
  global xsav1 tsav1 usav1;

  % Plot array as a mesh.
  mesh(xsav1, tsav1, usav1);

  % Label the axes, and set the title.
  xlabel('x');
  ylabel('t');
  zlabel('U(x,t)');
  title('Advection and Diffusion of a Cloud of Material');

  % Set the axes limits tight to the x and y range.
  axis([xsav1(1) xsav1(end) tsav1(1) tsav1(end) 0 1]);
  view(-12, 56);

function ex2_plot
  % Plot the results.
  global xsav2 tsav2 usav2;

  % Plot array as a mesh.
  mesh(xsav2, tsav2, usav2);

  % Label the axes, and set the title.
  xlabel('x');
  ylabel('t');
  zlabel('U(x,t)');
  title('Linear Advection with Non-linear Source Term');

  % Set the axes limits tight to the x and y range.
  axis([xsav2(1) xsav2(end) tsav2(1) tsav2(end) 0 1]);
  view(15, 30);

d03ps example results

 Example 1

 npts =   61 atol =  1.000e-04  rtol =  1.000e-04 

 t =  0.100
 x          0.2000   0.3000   0.4000   0.5000   0.6000   0.7000   0.8000
 approx u   0.0000   0.1198   0.9461   0.1182   0.0000   0.0000   0.0000

 t =  0.200
 x          0.2000   0.3000   0.4000   0.5000   0.6000   0.7000   0.8000
 approx u   0.0000   0.0007   0.1631   0.9015   0.1629   0.0001   0.0000

 t =  0.300
 x          0.2000   0.3000   0.4000   0.5000   0.6000   0.7000   0.8000
 approx u   0.0000   0.0000   0.0025   0.1924   0.8596   0.1946   0.0002

 Number of integration steps in time =     92
 Number of function evaluations =    443
 Number of Jacobian evaluations =     39
 Number of iterations =    231



 Example 2

 npts =   61 atol =  5.000e-04  rtol =  5.000e-02 

 t =  0.200
        x      approx u    exact u
   0.2000      1.0000      1.0000
   0.3000      0.9536      1.0000
   0.4000      0.0000      0.0000
   0.5000      0.0000      0.0000
   0.6000      0.0000      0.0000
   0.7000     -0.0000      0.0000
   0.8000      0.0000      0.0000


 t =  0.400
        x      approx u    exact u
   0.2000      1.0000      1.0000
   0.3000      1.0000      1.0000
   0.4000      1.0000      1.0000
   0.5000      0.9750      1.0000
   0.6000     -0.0000      0.0000
   0.7000      0.0000      0.0000
   0.8000      0.0000      0.0000


 Number of integration steps in time =    672
 Number of function evaluations =   1515
 Number of Jacobian evaluations =      1
 Number of iterations =      2