NAG FL Interface
d03pef (dim1_parab_keller)
1
Purpose
d03pef integrates a system of linear or nonlinear, firstorder, timedependent partial differential equations (PDEs) in one space variable. The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a Backward Differentiation Formula (BDF) method.
2
Specification
Fortran Interface
Subroutine d03pef ( 
npde, ts, tout, pdedef, bndary, u, npts, x, nleft, acc, rsave, lrsave, isave, lisave, itask, itrace, ind, ifail) 
Integer, Intent (In) 
:: 
npde, npts, nleft, lrsave, lisave, itask, itrace 
Integer, Intent (Inout) 
:: 
isave(lisave), ind, ifail 
Real (Kind=nag_wp), Intent (In) 
:: 
tout, x(npts), acc 
Real (Kind=nag_wp), Intent (Inout) 
:: 
ts, u(npde,npts), rsave(lrsave) 
External 
:: 
pdedef, bndary 

C Header Interface
#include <nag.h>
void 
d03pef_ (const Integer *npde, double *ts, const double *tout, void (NAG_CALL *pdedef)(const Integer *npde, const double *t, const double *x, const double u[], const double ut[], const double ux[], double res[], Integer *ires), void (NAG_CALL *bndary)(const Integer *npde, const double *t, const Integer *ibnd, const Integer *nobc, const double u[], const double ut[], double res[], Integer *ires), double u[], const Integer *npts, const double x[], const Integer *nleft, const double *acc, double rsave[], const Integer *lrsave, Integer isave[], const Integer *lisave, const Integer *itask, const Integer *itrace, Integer *ind, Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
d03pef_ (const Integer &npde, double &ts, const double &tout, void (NAG_CALL *pdedef)(const Integer &npde, const double &t, const double &x, const double u[], const double ut[], const double ux[], double res[], Integer &ires), void (NAG_CALL *bndary)(const Integer &npde, const double &t, const Integer &ibnd, const Integer &nobc, const double u[], const double ut[], double res[], Integer &ires), double u[], const Integer &npts, const double x[], const Integer &nleft, const double &acc, double rsave[], const Integer &lrsave, Integer isave[], const Integer &lisave, const Integer &itask, const Integer &itrace, Integer &ind, Integer &ifail) 
}

The routine may be called by the names d03pef or nagf_pde_dim1_parab_keller.
3
Description
d03pef integrates the system of firstorder PDEs
In particular the functions
${G}_{i}$ must have the general form
where
${P}_{i,j}$ and
${Q}_{i}$ depend on
$x$,
$t$,
$U$,
${U}_{x}$ and the vector
$U$ is the set of solution values
and the vector
${U}_{x}$ is its partial derivative with respect to
$x$. Note that
${P}_{i,j}$ and
${Q}_{i}$ must not depend on
$\frac{\partial U}{\partial t}$.
The integration in time is from ${t}_{0}$ to ${t}_{\mathrm{out}}$, over the space interval $a\le x\le b$, where $a={x}_{1}$ and $b={x}_{{\mathbf{npts}}}$ are the leftmost and rightmost points of a userdefined mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$. The mesh should be chosen in accordance with the expected behaviour of the solution.
The PDE system which is defined by the functions
${G}_{i}$ must be specified in
pdedef.
The initial values of the functions
$U\left(x,t\right)$ must be given at
$t={t}_{0}$. For a firstorder system of PDEs, only one boundary condition is required for each PDE component
${U}_{i}$. The
npde boundary conditions are separated into
${n}_{a}$ at the lefthand boundary
$x=a$, and
${n}_{b}$ at the righthand boundary
$x=b$, such that
${n}_{a}+{n}_{b}={\mathbf{npde}}$. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of
${U}_{i}$ at the lefthand boundary (say) points into the interior of the solution domain, then the boundary condition for
${U}_{i}$ should be specified at the lefthand boundary. Incorrect positioning of boundary conditions generally results in initialization or integration difficulties in the underlying time integration routines.
The boundary conditions have the form:
at the lefthand boundary, and
at the righthand boundary.
Note that the functions
${G}_{i}^{L}$ and
${G}_{i}^{R}$ must not depend on
${U}_{x}$, since spatial derivatives are not determined explicitly in the Keller box scheme (see
Keller (1970)). If the problem involves derivative (Neumann) boundary conditions then it is generally possible to restate such boundary conditions in terms of permissible variables. Also note that
${G}_{i}^{L}$ and
${G}_{i}^{R}$ must be linear with respect to time derivatives, so that the boundary conditions have the general form
at the lefthand boundary, and
at the righthand boundary, where
${E}_{i,j}^{L}$,
${E}_{i,j}^{R}$,
${S}_{i}^{L}$, and
${S}_{i}^{R}$ depend on
$x$,
$t$ and
$U$ only.
The boundary conditions must be specified in
bndary.
The problem is subject to the following restrictions:

(i)${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction;

(ii)${P}_{i,j}$ and ${Q}_{i}$ must not depend on any time derivatives;

(iii)The evaluation of the function ${G}_{i}$ is done at the midpoints of the mesh intervals by calling the pdedef for each midpoint in turn. Any discontinuities in the function must therefore be at one or more of the mesh points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$;

(iv)At least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the problem.
In this method of lines approach the Keller box scheme (see
Keller (1970)) is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of
${U}_{i}$ at each mesh point. In total there are
${\mathbf{npde}}\times {\mathbf{npts}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF method.
4
References
Berzins M (1990) Developments in the NAG Library software for parabolic equations Scientific Software Systems (eds J C Mason and M G Cox) 59–72 Chapman and Hall
Berzins M, Dew P M and Furzeland R M (1989) Developing software for timedependent problems using the method of lines and differentialalgebraic integrators Appl. Numer. Math. 5 375–397
Keller H B (1970) A new difference scheme for parabolic problems Numerical Solutions of Partial Differential Equations (ed J Bramble) 2 327–350 Academic Press
Pennington S V and Berzins M (1994) New NAG Library software for firstorder partial differential equations ACM Trans. Math. Softw. 20 63–99
5
Arguments

1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system to be solved.
Constraint:
${\mathbf{npde}}\ge 1$.

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

On entry: the initial value of the independent variable $t$.
Constraint:
${\mathbf{ts}}<{\mathbf{tout}}$.
On exit: the value of
$t$ corresponding to the solution values in
u. Normally
${\mathbf{ts}}={\mathbf{tout}}$.

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

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

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

pdedef must 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:
Fortran Interface
Integer, Intent (In) 
:: 
npde 
Integer, Intent (Inout) 
:: 
ires 
Real (Kind=nag_wp), Intent (In) 
:: 
t, x, u(npde), ut(npde), ux(npde) 
Real (Kind=nag_wp), Intent (Out) 
:: 
res(npde) 

C Header Interface
void 
pdedef_ (const Integer *npde, const double *t, const double *x, const double u[], const double ut[], const double ux[], double res[], Integer *ires) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
pdedef_ (const Integer &npde, const double &t, const double &x, const double u[], const double ut[], const double ux[], double res[], Integer &ires) 
}


1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – Real (Kind=nag_wp)
Input

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

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

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

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

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

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

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

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

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

7:
$\mathbf{res}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) array
Output

On exit:
${\mathbf{res}}\left(\mathit{i}\right)$ must contain the
$\mathit{i}$th component of
$G$, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, where
$G$ is defined as
i.e., only terms depending explicitly on time derivatives, or
i.e., all terms in equation
(2).
The definition of
$G$ is determined by the input value of
ires.

8:
$\mathbf{ires}$ – Integer
Input/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 ${\mathbf{ires}}=3$, 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. Arguments denoted as
Input must
not be changed by this procedure.
Note: pdedef should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pef. If your code inadvertently
does return any NaNs or infinities,
d03pef is likely to produce unexpected results.

5:
$\mathbf{bndary}$ – 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:
Fortran Interface
Integer, Intent (In) 
:: 
npde, ibnd, nobc 
Integer, Intent (Inout) 
:: 
ires 
Real (Kind=nag_wp), Intent (In) 
:: 
t, u(npde), ut(npde) 
Real (Kind=nag_wp), Intent (Out) 
:: 
res(nobc) 

C Header Interface
void 
bndary_ (const Integer *npde, const double *t, const Integer *ibnd, const Integer *nobc, const double u[], const double ut[], double res[], Integer *ires) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
bndary_ (const Integer &npde, const double &t, const Integer &ibnd, const Integer &nobc, const double u[], const double ut[], double res[], Integer &ires) 
}


1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – Real (Kind=nag_wp)
Input

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

3:
$\mathbf{ibnd}$ – Integer
Input

On entry: determines the position of the boundary conditions.
 ${\mathbf{ibnd}}=0$
 bndary must compute the lefthand boundary condition at $x=a$.
 ${\mathbf{ibnd}}\ne 0$
 Indicates that bndary must compute the righthand boundary condition at $x=b$.

4:
$\mathbf{nobc}$ – Integer
Input

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

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

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

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

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

7:
$\mathbf{res}\left({\mathbf{nobc}}\right)$ – Real (Kind=nag_wp) array
Output

On exit:
${\mathbf{res}}\left(\mathit{i}\right)$ must contain the
$\mathit{i}$th component of
${G}^{L}$ or
${G}^{R}$, depending on the value of
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{nobc}}$, where
${G}^{L}$ is defined as
i.e., only terms depending explicitly on time derivatives, or
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:
$\mathbf{ires}$ – Integer
Input/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 ${\mathbf{ires}}=3$, 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. Arguments denoted as
Input must
not be changed by this procedure.
Note: bndary should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pef. If your code inadvertently
does return any NaNs or infinities,
d03pef is likely to produce unexpected results.

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

On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint:
${\mathbf{npts}}\ge 3$.

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

On entry: the mesh points in the spatial direction. ${\mathbf{x}}\left(1\right)$ must specify the lefthand boundary, $a$, and ${\mathbf{x}}\left({\mathbf{npts}}\right)$ must specify the righthand boundary, $b$.
Constraint:
${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{npts}}\right)$.

9:
$\mathbf{nleft}$ – Integer
Input

On entry: the number ${n}_{a}$ of boundary conditions at the lefthand mesh point ${\mathbf{x}}\left(1\right)$.
Constraint:
$0\le {\mathbf{nleft}}\le {\mathbf{npde}}$.

10:
$\mathbf{acc}$ – 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:
Constraint:
${\mathbf{acc}}>0.0$.

11:
$\mathbf{rsave}\left({\mathbf{lrsave}}\right)$ – Real (Kind=nag_wp) array
Communication Array

If
${\mathbf{ind}}=0$,
rsave need not be set on entry.
If
${\mathbf{ind}}=1$,
rsave must be unchanged from the previous call to the routine because it contains required information about the iteration.

12:
$\mathbf{lrsave}$ – Integer
Input

On entry: the dimension of the array
rsave as declared in the (sub)program from which
d03pef is called.
Constraint:
${\mathbf{lrsave}}\ge \left(4\times {\mathbf{npde}}+{\mathbf{nleft}}+14\right)\times {\mathbf{npde}}\times {\mathbf{npts}}+\left(3\times {\mathbf{npde}}+21\right)\times {\mathbf{npde}}+\phantom{\rule{0ex}{0ex}}7\times {\mathbf{npts}}+54$.

13:
$\mathbf{isave}\left({\mathbf{lisave}}\right)$ – Integer array
Communication 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 backsubstitution using the $LU$ decomposition of the Jacobian matrix.

14:
$\mathbf{lisave}$ – Integer
Input

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:
$\mathbf{itask}$ – Integer
Input

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:
$\mathbf{itrace}$ – Integer
Input

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
Subchapter D02MN.

17:
$\mathbf{ind}$ – Integer
Input/Output

On entry: indicates whether this is a continuation call or a new integration.
 ${\mathbf{ind}}=0$
 Starts or restarts the integration in time.
 ${\mathbf{ind}}=1$
 Continues the integration after an earlier exit from the routine. In this case, only the arguments tout and ifail should be reset between calls to d03pef.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.

18:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$0$ is recommended.
When the value $\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{acc}}>0.0$.
On entry, $\mathit{i}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{x}}\left(\mathit{i}\right)=\u2329\mathit{\text{value}}\u232a$, $\mathit{j}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{x}}\left(\mathit{j}\right)=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{npts}}\right)$.
On entry, ${\mathbf{ind}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{itask}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{itask}}=1$, $2$ or $3$.
On entry, ${\mathbf{lisave}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lisave}}\ge \u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{lrsave}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lrsave}}\ge \u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{nleft}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{npde}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nleft}}\le {\mathbf{npde}}$.
On entry, ${\mathbf{nleft}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nleft}}\ge 0$.
On entry, ${\mathbf{npde}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npts}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npts}}\ge 3$.
On entry, on initial entry ${\mathbf{ind}}=1$.
Constraint: on initial entry ${\mathbf{ind}}=0$.
On entry, ${\mathbf{tout}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tout}}>{\mathbf{ts}}$.
On entry, ${\mathbf{tout}}{\mathbf{ts}}$ is too small:
${\mathbf{tout}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=2$

Underlying ODE solver cannot make further progress from the point
ts with the supplied value of
acc.
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=3$

Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
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 an internal auxiliary was unable to initialize the derivative. This could be due to your setting
${\mathbf{ires}}=3$ in
pdedef or
bndary.
 ${\mathbf{ifail}}=5$

Singular Jacobian of ODE system. Check problem formulation.
 ${\mathbf{ifail}}=6$

In evaluating residual of ODE system,
${\mathbf{ires}}=2$ has been set in
pdedef or
bndary. Integration is successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=7$

acc was too small to start integration:
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=8$

ires set to an invalid value in call to
pdedef or
bndary.
 ${\mathbf{ifail}}=9$

Serious error in internal call to an auxiliary. Increase
itrace for further details.
 ${\mathbf{ifail}}=10$

Integration completed, but a small change in
acc is unlikely to result in a changed solution.
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
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
${\mathbf{itask}}\ne 2$.)
 ${\mathbf{ifail}}=11$

Error during Jacobian formulation for ODE system. Increase
itrace for further details.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
d03pef controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy argument,
acc.
8
Parallelism and Performance
d03pef is not thread safe and should not be called from a multithreaded user program. Please see
Section 1 in FL Interface Multithreading for more information on thread safety.
d03pef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The Keller box scheme can be used to solve higherorder problems which have been reduced to firstorder by the introduction of new variables (see the example problem in
d03pkf). In general, a secondorder problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (
d03pcf/d03pca or
d03phf/d03pha for example), but at the expense of increased CPU time due to the larger number of function evaluations required.
It should be noted that the Keller box scheme, in common with other centraldifference schemes, may be unsuitable for some hyperbolic firstorder problems such as the apparently simple linear advection equation
${U}_{t}+a{U}_{x}=0$, where
$a$ is a constant, resulting in spurious oscillations due to the lack of dissipation. This type of problem requires a discretization scheme with upwind weighting (
d03pff for example), or the addition of a secondorder artificial dissipation term.
The time taken depends on the complexity of the system and on the accuracy requested.
10
Example
This example is the simple firstorder system
for
$t\in \left[0,1\right]$ and
$x\in \left[0,1\right]$.
The initial conditions are
and the Dirichlet boundary conditions for
${U}_{1}$ at
$x=0$ and
${U}_{2}$ at
$x=1$ are given by the exact solution:
10.1
Program Text
10.2
Program Data
10.3
Program Results