d03pzf interpolates in the spatial coordinate the solution and derivative of a system of partial differential equations (PDEs). The solution must first be computed using one of the finite difference schemes d03pcf/d03pca, d03phf/d03pha or d03ppf/d03ppa, or one of the Keller box schemes d03pef, d03pkf or d03prf.

2 Specification

Fortran Interface

Subroutine d03pzf (

npde, m, u, npts, x, xp, intpts, itype, up, ifail)

Integer, Intent (In)	::	npde, m, npts, intpts, itype
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	u(npde,npts), x(npts), xp(intpts)
Real (Kind=nag_wp), Intent (Out)	::	up(npde,intpts,itype)

C Header Interface

#include <nag.h>

void	d03pzf_ (const Integer npde, const Integer m, const double u[], const Integer npts, const double x[], const double xp[], const Integer intpts, const Integer itype, double up[], Integer ifail)

The routine may be called by the names d03pzf or nagf_pde_dim1_parab_fd_interp.

3 Description

d03pzf is an interpolation routine for evaluating the solution of a system of partial differential equations (PDEs), at a set of user-specified points. The solution of the system of equations (possibly with coupled ordinary differential equations) must be computed using a finite difference scheme or a Keller box scheme on a set of mesh points. d03pzf can then be employed to compute the solution at a set of points anywhere in the range of the mesh. It can also evaluate the first spatial derivative of the solution. It uses linear interpolation for approximating the solution.

4 References

None.

5 Arguments

Note: the arguments x, m, u, npts and npde must be supplied unchanged from the PDE routine.

1: $npde$ – Integer Input

On entry: the number of PDEs.

Constraint:

npde \geq 1

2: $m$ – Integer Input

On entry: the coordinate system used. If the call to d03pzf follows one of the finite difference routines then m must be the same argument m as used in that call. For the Keller box scheme only Cartesian coordinate systems are valid and so m must be set to zero. No check will be made by d03pzf in this case.

$m = 0$: Indicates Cartesian coordinates.
$m = 1$: Indicates cylindrical polar coordinates.
$m = 2$: Indicates spherical polar coordinates.

Constraints:

$0 \leq m \leq 2$ following a finite difference routine;
$m = 0$ following a Keller box scheme routine.

3: $u (npde, npts)$ – Real (Kind=nag_wp) array Input

On entry: the PDE part of the original solution returned in the argument u by the PDE routine.

Constraint:

npde \geq 1

4: $npts$ – Integer Input

On entry: the number of mesh points.

Constraint:

npts \geq 3

5: $x (npts)$ – Real (Kind=nag_wp) array Input

On entry:

x (i)

, for

i = 1, 2, \dots, npts

, must contain the mesh points as used by the PDE routine.

6: $xp (intpts)$ – Real (Kind=nag_wp) array Input

On entry:

xp (i)

, for

i = 1, 2, \dots, intpts

, must contain the spatial interpolation points.

Constraint:

x (1) \leq xp (1) < xp (2) < \dots < xp (intpts) \leq x (npts)

7: $intpts$ – Integer Input

On entry: the number of interpolation points.

Constraint:

intpts \geq 1

8: $itype$ – Integer Input

On entry: specifies the interpolation to be performed.

$itype = 1$: The solutions at the interpolation points are computed.
$itype = 2$: Both the solutions and their first derivatives at the interpolation points are computed.

Constraint:

itype = 1

2

9: $up (npde, intpts, itype)$ – Real (Kind=nag_wp) array Output

On exit: if

itype = 1

up (i, j, 1)

, contains the value of the solution

U_{i} (x_{j}, t_{out})

, at the interpolation points

x_{j} = xp (j)

, for

j = 1, 2, \dots, intpts

and

i = 1, 2, \dots, npde

itype = 2

up (i, j, 1)

contains

U_{i} (x_{j}, t_{out})

and

up (i, j, 2)

contains

\frac{\partial U_{i}}{\partial x}

at these points.

10: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $i = ⟨ value ⟩$ , $x (i) = ⟨ value ⟩$ , $j = ⟨ value ⟩$ and $x (j) = ⟨ value ⟩$ .
Constraint: $x (1) < x (2) < \dots < x (npts)$ .

On entry, $intpts = ⟨ value ⟩$ .
Constraint: $intpts \geq 1$ .

On entry, $itype = ⟨ value ⟩$ .
Constraint: $itype = 1$ or $2$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m = 0$ , $1$ or $2$ .

On entry, $npde = ⟨ value ⟩$ .
Constraint: $npde > 0$ .

On entry, $npts = ⟨ value ⟩$ .
Constraint: $npts > 2$ .

$ifail = 2$: On entry, $i = ⟨ value ⟩$ , $xp (i) = ⟨ value ⟩$ , $j = ⟨ value ⟩$ and $xp (j) = ⟨ value ⟩$ .
Constraint: $x (1) \leq xp (1) < xp (2) < \dots < xp (intpts) \leq x (npts)$ .

$ifail = 3$: On entry, interpolating point $⟨ value ⟩$ with the value $⟨ value ⟩$ is outside the x range.

$ifail = - 99$: 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.

$ifail = - 399$: 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.