NAG CL Interface
d03pec (dim1_parab_keller)
1
Purpose
d03pec 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
void 
d03pec (Integer npde,
double *ts,
double tout,
void 
(*pdedef)(Integer npde,
double t,
double x,
const double u[],
const double ut[],
const double ux[],
double res[],
Integer *ires,
Nag_Comm *comm),


double u[],
Integer npts,
const double x[],
Integer nleft,
double acc,
double rsave[],
Integer lrsave,
Integer isave[],
Integer lisave,
Integer itask,
Integer itrace,
const char *outfile,
Integer *ind,
Nag_Comm *comm, Nag_D03_Save *saved,
NagError *fail) 

The function may be called by the names: d03pec, nag_pde_dim1_parab_keller or nag_pde_parab_1d_keller.
3
Description
d03pec 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 functions.
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:
$\mathbf{ts}$ – double *
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:
$\mathbf{tout}$ – double
Input

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

4:
$\mathbf{pdedef}$ – function, supplied by the user
External Function

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
d03pec.
The specification of
pdedef is:
void 
pdedef (Integer npde,
double t,
double x,
const double u[],
const double ut[],
const double ux[],
double res[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
Input

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

3:
$\mathbf{x}$ – double
Input

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

4:
$\mathbf{u}\left[{\mathbf{npde}}\right]$ – const double
Input

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

5:
$\mathbf{ut}\left[{\mathbf{npde}}\right]$ – const double
Input

On entry: ${\mathbf{ut}}\left[\mathit{i}1\right]$ 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:
$\mathbf{ux}\left[{\mathbf{npde}}\right]$ – const double
Input

On entry: ${\mathbf{ux}}\left[\mathit{i}1\right]$ 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:
$\mathbf{res}\left[{\mathbf{npde}}\right]$ – double
Output

On exit:
${\mathbf{res}}\left[\mathit{i}1\right]$ 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 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 function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 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 ${\mathbf{ires}}=3$, d03pec returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

9:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
pdedef.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03pec you may allocate memory and initialize these pointers with various quantities for use by
pdedef when called from
d03pec (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: pdedef should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pec. If your code inadvertently
does return any NaNs or infinities,
d03pec is likely to produce unexpected results.

5:
$\mathbf{bndary}$ – function, supplied by the user
External Function

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:
void 
bndary (Integer npde,
double t,
Integer ibnd,
Integer nobc,
const double u[],
const double ut[],
double res[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
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:
$\mathbf{u}\left[{\mathbf{npde}}\right]$ – const double
Input

On entry:
${\mathbf{u}}\left[\mathit{i}1\right]$ 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:
$\mathbf{ut}\left[{\mathbf{npde}}\right]$ – const double
Input

On entry:
${\mathbf{ut}}\left[\mathit{i}1\right]$ 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:
$\mathbf{res}\left[\mathit{dim}\right]$ – double
Output

On exit:
${\mathbf{res}}\left[\mathit{i}1\right]$ 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 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 function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 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 ${\mathbf{ires}}=3$, d03pec returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

9:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
bndary.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03pec you may allocate memory and initialize these pointers with various quantities for use by
bndary when called from
d03pec (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: bndary should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pec. If your code inadvertently
does return any NaNs or infinities,
d03pec is likely to produce unexpected results.

6:
$\mathbf{u}\left[{\mathbf{npde}}\times {\mathbf{npts}}\right]$ – double
Input/Output

Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: the initial values of $U\left(x,t\right)$ at $t={\mathbf{ts}}$ and the mesh points
${\mathbf{x}}\left[\mathit{j}1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
On exit: ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ 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:
$\mathbf{x}\left[{\mathbf{npts}}\right]$ – const double
Input

On entry: the mesh points in the spatial direction. ${\mathbf{x}}\left[0\right]$ must specify the lefthand boundary, $a$, and ${\mathbf{x}}\left[{\mathbf{npts}}1\right]$ must specify the righthand boundary, $b$.
Constraint:
${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}1\right]$.

9:
$\mathbf{nleft}$ – Integer
Input

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

10:
$\mathbf{acc}$ – double
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:
$\mathbf{rsave}\left[{\mathbf{lrsave}}\right]$ – double
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 function because it contains required information about the iteration.

12:
$\mathbf{lrsave}$ – Integer
Input

On entry: the dimension of the array
rsave.
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
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 function because it contains required information about the iteration. In particular:
 ${\mathbf{isave}}\left[0\right]$
 Contains the number of steps taken in time.
 ${\mathbf{isave}}\left[1\right]$
 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.
 ${\mathbf{isave}}\left[2\right]$
 Contains the number of Jacobian evaluations performed by the time integrator.
 ${\mathbf{isave}}\left[3\right]$
 Contains the order of the last backward differentiation formula method used.
 ${\mathbf{isave}}\left[4\right]$
 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.
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
d03pec 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.
 ${\mathbf{itrace}}=1$
 Output from the underlying ODE solver is printed. 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 ${\mathbf{itrace}}=0$.

17:
$\mathbf{outfile}$ – const char *
Input

On entry: the name of a file to which diagnostic output will be directed. If
outfile is
NULL the diagnostic output will be directed to standard output.

18:
$\mathbf{ind}$ – 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 function. In this case, only the argument tout should be reset between calls to d03pec.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.

19:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

20:
$\mathbf{saved}$ – Nag_D03_Save *
Communication Structure

saved must remain unchanged following a previous call to a
Chapter D03 function and prior to any subsequent call to a
Chapter D03 function.

21:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ACC_IN_DOUBT

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
${\mathbf{itask}}\ne 2$.)
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_FAILED_DERIV

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.
 NE_FAILED_START

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

Error during Jacobian formulation for ODE system. Increase
itrace for further details.
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
$t={\mathbf{ts}}$.
Underlying ODE solver cannot make further progress from the point
ts with the supplied value of
acc.
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INT

ires set to an invalid value in call to
pdedef or
bndary.
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, ${\mathbf{nleft}}=\u2329\mathit{\text{value}}\u232a$.
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 ${\mathbf{ind}}=0$.
 NE_INT_2

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: ${\mathbf{nleft}}\le {\mathbf{npde}}$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Serious error in internal call to an auxiliary. Increase
itrace for further details.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_NOT_CLOSE_FILE

Cannot close file $\u2329\mathit{\text{value}}\u232a$.
 NE_NOT_STRICTLY_INCREASING

On entry, $\mathit{i}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{x}}\left[\mathit{i}1\right]=\u2329\mathit{\text{value}}\u232a$, $\mathit{j}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{x}}\left[\mathit{j}1\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}1\right]$.
 NE_NOT_WRITE_FILE

Cannot open file $\u2329\mathit{\text{value}}\u232a$ for writing.
 NE_REAL

On entry, ${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{acc}}>0.0$.
 NE_REAL_2

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 ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 NE_SING_JAC

Singular Jacobian of ODE system. Check problem formulation.
 NE_USER_STOP

In evaluating residual of ODE system,
${\mathbf{ires}}=2$ has been set in
pdedef or
bndary. Integration is successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
d03pec 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
d03pec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pec 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 function. 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
d03pkc). In general, a secondorder problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (
d03pcc or
d03phc 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 (
d03pfc 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
None.
10.3
Program Results