NAG Toolbox: nag_pde_1d_parab_keller (d03pe)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{ts}$ – double scalar

The initial value of the independent variable $t$.

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

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

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

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

pdedef must compute the functions $G_i$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_1d_parab_keller (d03pe).

[res, ires] = pdedef(npde, t, x, u, ut, ux, 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:     $\mathrm{ut}\left(\mathbf{npde}\right)$ – double array

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

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

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

The form of $G_i$ that must be returned in the array res.

$\mathbf{ires}=-1$

Equation (8) must be used.

$\mathbf{ires}=1$

Equation (9) must be used.

$\mathbf{ires}=2$

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_keller (d03pe) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

Output Parameters

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

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

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}\text{,}$$

(8)

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

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}+Q_{\mathit{i}}\text{,}$$

(9)

i.e., all terms in equation (2).

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

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_keller (d03pe) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

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

bndary must compute the functions $G_i^L$ and $G_i^R$ which define the boundary conditions as in equations (4) and (5).

[res, ires] = bndary(npde, t, ibnd, nobc, u, ut, 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:     $\mathrm{ibnd}$ – int64int32nag_int scalar

Determines the position of the boundary conditions.

$\mathbf{ibnd}=0$

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

$\mathbf{ibnd}\ne 0$

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

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

Specifies the number of boundary conditions at the boundary specified by ibnd.

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

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

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

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

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

The form $G_i^L$ (or $G_i^R$) that must be returned in the array res.

$\mathbf{ires}=-1$

Equation (10) must be used.

$\mathbf{ires}=1$

Equation (11) must be used.

Output Parameters

1:     $\mathrm{res}\left(\mathbf{nobc}\right)$ – double array

$\mathbf{res}\left(\mathit{i}\right)$ 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

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}\text{,}$$

(10)

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

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

(11)

i.e., all terms in equation (6), and similarly for $G_{\mathit{i}}^R$.
The definitions of $G^L$ and $G^R$ are determined by the input value of ires.

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_keller (d03pe) returns to the calling function with the error indicator set to $\mathbf{ifail}=\mathbf{4}$.

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

The initial values of $U\left(x,t\right)$ at $t=\mathbf{ts}$ and the mesh points $\mathbf{x}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{npts}$.

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

The mesh points in the spatial 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: $\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{npts}\right)$.

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

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

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

8:     $\mathrm{acc}$ – double scalar

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

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

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

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

lrsave, the dimension of the array, must satisfy the constraint $$\begin{gathered} \mathit{lrsave}\ge \left(4\times \mathbf{npde}+\mathbf{nleft}+14\right)\times \mathbf{npde}\times \mathbf{npts}+\left(3\times \mathbf{npde}+21\right)\times \mathbf{npde}+ \\ 7\times \mathbf{npts}+54 \end{gathered}$$.

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.

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

lisave, the dimension of the array, must satisfy the constraint $\mathit{lisave}\ge \mathbf{npde}\times \mathbf{npts}+24$.

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

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

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

Contains the number of steps taken in time.

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

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

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

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

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

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

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

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

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

Specifies the task to be performed by the ODE integrator.

$\mathbf{itask}=1$

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

$\mathbf{itask}=2$

Take one step and return.

$\mathbf{itask}=3$

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

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

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

The level of trace information required from nag_pde_1d_parab_keller (d03pe) 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 nag_file_set_unit_error (x04aa)).

$\mathbf{itrace}=1$

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.

$\mathbf{itrace}=2$

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

$\mathbf{itrace}\ge 3$

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

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

13:   $\mathrm{ind}$ – 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 and ifail should be reset between calls to nag_pde_1d_parab_keller (d03pe).

Constraint: $\mathbf{ind}=0$ or $1$.

Optional Input Parameters

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

Default: the first dimension of the array u.

The number of PDEs in the system to be solved.

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

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

Default: the dimension of the array x and the second dimension of the array u. (An error is raised if these dimensions are not equal.)

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

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

Output Parameters

1:     $\mathrm{ts}$ – double scalar

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

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

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

3:     $\mathrm{rsave}\left(\mathit{lrsave}\right)$ – 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.

4:     $\mathrm{isave}\left(\mathit{lisave}\right)$ – 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:

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

Contains the number of steps taken in time.

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

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

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

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

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

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

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

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

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

$\mathbf{ind}=1$.

6:     $\mathrm{ifail}$ – 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,	$\mathbf{tout}\le \mathbf{ts}$,
or	$\left(\mathbf{tout}-\mathbf{ts}\right)$ is too small,
or	$\mathbf{itask}\ne 1$, $2$ or $3$,
or	$\text{mesh points }\mathbf{x}\left(i\right)$ are not ordered correctly,
or	$\mathbf{npts}<3$,
or	$\mathbf{npde}<1$,
or	nleft is not in the range $0$ to npde,
or	$\mathbf{acc}\le 0.0$,
or	$\mathbf{ind}\ne 0$ or $1$,
or	lrsave is too small,
or	lisave is too small,
or	nag_pde_1d_parab_keller (d03pe) called initially with $\mathbf{ind}=1$.

W  $\mathbf{ifail}=2$

The underlying ODE solver cannot make any further progress across the integration range from the current point $t=\mathbf{ts}$ with the supplied value of acc. 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 errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or acc is too small for the integration to continue. Incorrect positioning of boundary conditions may also result in this error. Integration was successful as far as $t=\mathbf{ts}$.

   $\mathbf{ifail}=4$

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 the pdedef or bndary, when the residual in the underlying ODE solver was being evaluated. Incorrect positioning of boundary conditions may also result in this error.

   $\mathbf{ifail}=5$

In solving the ODE system, a singular Jacobian has been encountered. You should check their problem formulation.

W  $\mathbf{ifail}=6$

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

   $\mathbf{ifail}=7$

The value of acc is so small that the function is unable to start the integration in time.

   $\mathbf{ifail}=8$

In either, pdedef or bndary, 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 acc is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $\mathbf{itask}\ne 2$.)

   $\mathbf{ifail}=11$

An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current advisory message unit).

   $\mathbf{ifail}=-99$

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: d03pe_example

function d03pe_example


fprintf('d03pe example results\n\n');

% Initial conditions
ts    = 0;
dx    = 0.025;
x     = [0:dx:1];
npts  = 1 + 1/0.025;
u(1,1:npts) = exp(x);
u(2,1:npts) = sin(x);

% Input parameter values
nleft = int64(1);
acc   = 1e-06;
rsave = zeros(4000, 1);
isave = zeros(200, 1, 'int64');
itask = int64(1);
itrace = int64(0);
ind   = int64(0);

% Solve system at time-steps of 0.2
fprintf('%32s\n%6s    ','x','t');
fprintf('%10.4f',x(5:8:37));
fprintf('\n');

pr = false;
i = 0;
t = [0.1:0.1:1];
for tout  = t
  i = i+1;
  [ts, u, rsave, isave, ind, ifail] = ...
    d03pe(...
          ts, tout, @pdedef, @bndary, u, x, nleft, acc, ...
          rsave, isave, itask, itrace, ind);
  if pr
    fprintf('%6.1f %3s',ts,'U1:');
    fprintf('%10.4f',u(1,5:8:37));
    fprintf('\n%10s','U2:');
    fprintf('%10.4f',u(2,5:8:37));
    fprintf('\n');
  end
  pr = ~pr;
  v(:,i) = u(1,:);
  w(:,i) = u(2,:);
end

fig1 = figure;
hold on
mesh(t,x,v);
mesh(t,x,w);
xlabel('Time');
ylabel('x');
title('First-order solution using Keller box and DBF');
legend('u(1,x,t)','u(2,x,t)');
view(47,26);
hold off



function [res, ires] = pdedef(npde, t, x, u, ut, ux, ires)
  if (ires == -1)
    res(1:2) = ut(1:2);
  else
    res(1) = ut(1) +   ux(1) + ux(2);
    res(2) = ut(2) + 4*ux(1) + ux(2);
  end

function [res, ires] = bndary(npde, t, ibnd, nobc, u, ut, ires)
  if (ibnd == 0)
    if (ires == -1)
      res(1) = 0;
    else
      res(1) = u(1) - (exp(t)+exp(-3*t))/2 - (sin(-3*t)-sin(t))/4;
    end
  else
    if (ires == -1)
      res(1) = 0;
    else
      res(1) = u(2) - exp(1-3*t) + exp(1+t) - (sin(1-3*t)+sin(1+t))/2;
    end
  end

d03pe example results

                               x
     t        0.1000    0.3000    0.5000    0.7000    0.9000
   0.2 U1:    0.7845    1.0010    1.2733    1.6115    2.0281
       U2:   -0.8352   -0.8159   -0.8367   -0.9128   -1.0609
   0.4 U1:    0.6481    0.8533    1.1212    1.4627    1.8903
       U2:   -1.5216   -1.6767   -1.8934   -2.1917   -2.5945
   0.6 U1:    0.6892    0.8961    1.1747    1.5374    1.9989
       U2:   -2.0047   -2.3434   -2.7677   -3.3002   -3.9680
   0.8 U1:    0.8977    1.1247    1.4320    1.8349    2.3513
       U2:   -2.3403   -2.8675   -3.5110   -4.2960   -5.2537
   1.0 U1:    1.2470    1.5205    1.8829    2.3528    2.9519
       U2:   -2.6229   -3.3338   -4.1999   -5.2506   -6.5218