NAG Library Routine Document

D03PEF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NPDE – INTEGERInput

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

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

2:     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:     TOUT – REAL (KIND=nag_wp)Input

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

4:     PDEDEF – SUBROUTINE, supplied by the user.External Procedure

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 D03PEF.

The specification of PDEDEF is:

SUBROUTINE PDEDEF ( NPDE, T, X, U, UT, UX, RES, IRES) INTEGER  NPDE, IRES REAL (KIND=nag_wp)  T, X, U(NPDE), UT(NPDE), UX(NPDE), RES(NPDE)

1:     NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2:     T – REAL (KIND=nag_wp)Input

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

3:     X – REAL (KIND=nag_wp)Input

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

4:     U(NPDE) – REAL (KIND=nag_wp) arrayInput

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

5:     UT(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{UT}\left(\mathit{i}\right)$ 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:     UX(NPDE) – REAL (KIND=nag_wp) arrayInput

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

7:     RES(NPDE) – REAL (KIND=nag_wp) arrayOutput

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

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

(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.

8:     IRES – INTEGERInput/Output

On entry: 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.

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 $\mathbf{IRES}=3$, then D03PEF 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 D03PEF is called. Parameters denoted as Input must not be changed by this procedure.

5:     BNDARY – SUBROUTINE, supplied by the user.External Procedure

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

The specification of BNDARY is:

SUBROUTINE BNDARY ( NPDE, T, IBND, NOBC, U, UT, RES, IRES) INTEGER  NPDE, IBND, NOBC, IRES REAL (KIND=nag_wp)  T, U(NPDE), UT(NPDE), RES(NOBC)

1:     NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2:     T – REAL (KIND=nag_wp)Input

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

3:     IBND – INTEGERInput

On entry: 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:     NOBC – INTEGERInput

On entry: specifies the number of boundary conditions at the boundary specified by IBND.

5:     U(NPDE) – REAL (KIND=nag_wp) arrayInput

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

6:     UT(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{UT}\left(\mathit{i}\right)$ 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:     RES(NOBC) – REAL (KIND=nag_wp) arrayOutput

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

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

(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.

8:     IRES – INTEGERInput/Output

On entry: 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.

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 $\mathbf{IRES}=3$, then D03PEF 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 D03PEF is called. Parameters denoted as Input must not be changed by this procedure.

6:     U(NPDE,NPTS) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the initial values of $U\left(x,t\right)$ at $t=\mathbf{TS}$ and the mesh points $\mathbf{X}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{NPTS}$.

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

7:     NPTS – INTEGERInput

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

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

8:     X(NPTS) – REAL (KIND=nag_wp) arrayInput

On entry: 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: $\mathbf{X}\left(1\right)<\mathbf{X}\left(2\right)<\cdots <\mathbf{X}\left(\mathbf{NPTS}\right)$.

9:     NLEFT – INTEGERInput

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

Constraint: $0\le \mathbf{NLEFT}\le \mathbf{NPDE}$.

10:   ACC – REAL (KIND=nag_wp)Input

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

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

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

11:   RSAVE(LRSAVE) – REAL (KIND=nag_wp) arrayCommunication Array

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

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

12:   LRSAVE – INTEGERInput

On entry: the dimension of the array RSAVE as declared in the (sub)program from which D03PEF is called.

Constraint: $$\begin{gathered} \mathbf{LRSAVE}\ge \left(4\times \mathbf{NPDE}+\mathbf{NLEFT}+14\right)\times \mathbf{NPDE}\times \mathbf{NPTS}+\left(3\times \mathbf{NPDE}+21\right)\times \mathbf{NPDE}+ \\ 7\times \mathbf{NPTS}+54 \end{gathered}$$.

13:   ISAVE(LISAVE) – INTEGER arrayCommunication Array

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

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

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

Contains the number of steps taken in time.

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

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

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

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

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

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

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

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

14:   LISAVE – INTEGERInput

On entry: the dimension of the array ISAVE as declared in the (sub)program from which D03PEF is called.

Constraint: $\mathbf{LISAVE}\ge \mathbf{NPDE}\times \mathbf{NPTS}+24$.

15:   ITASK – INTEGERInput

On entry: 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 $3$.

16:   ITRACE – INTEGERInput

On entry: the level of trace information required from D03PEF 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.

$\mathbf{ITRACE}\ge 3$

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.

17:   IND – INTEGERInput/Output

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

$\mathbf{IND}=0$

Starts or restarts the integration in time.

$\mathbf{IND}=1$

Continues the integration after an earlier exit from the routine. In this case, only the parameters TOUT and IFAIL should be reset between calls to D03PEF.

Constraint: $\mathbf{IND}=0$ or $1$.

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

18:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

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

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

6  Error Indicators and 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	D03PEF called initially with $\mathbf{IND}=1$.

$\mathbf{IFAIL}=2$

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 $t=\mathbf{TS}$.

$\mathbf{IFAIL}=3$

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 routine 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.

$\mathbf{IFAIL}=6$

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 routine is unable to start the integration in time.

$\mathbf{IFAIL}=8$

In either, PDEDEF or BNDARY, IRES was set to an invalid value.

$\mathbf{IFAIL}=9$ (D02NNF)

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

$\mathbf{IFAIL}=10$

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).

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d03pefe.f90)

9.2  Program Data

Program Data (d03pefe.d)

9.3  Program Results

Program Results (d03pefe.r)