d03 Chapter Contents
d03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_pde_interp_1d_fd (d03pzc)

## 1  Purpose

nag_pde_interp_1d_fd (d03pzc) 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 nag_pde_parab_1d_fd (d03pcc), nag_pde_parab_1d_fd_ode (d03phc) or nag_pde_parab_1d_fd_ode_remesh (d03ppc), or one of the Keller box schemes nag_pde_parab_1d_keller (d03pec), nag_pde_parab_1d_keller_ode (d03pkc) or nag_pde_parab_1d_keller_ode_remesh (d03prc).

## 2  Specification

 #include #include
 void nag_pde_interp_1d_fd (Integer npde, Integer m, const double u[], Integer npts, const double x[], const double xp[], Integer intpts, Integer itype, double up[], NagError *fail)

## 3  Description

nag_pde_interp_1d_fd (d03pzc) is an interpolation function 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. nag_pde_interp_1d_fd (d03pzc) 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.

None.

## 5  Arguments

Note: the arguments x, m, u, npts and npde must be supplied unchanged from the PDE function.
1:     npdeIntegerInput
On entry: the number of PDEs.
Constraint: ${\mathbf{npde}}\ge 1$.
2:     mIntegerInput
On entry: the coordinate system used. If the call to nag_pde_interp_1d_fd (d03pzc) follows one of the finite difference functions 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 nag_pde_interp_1d_fd (d03pzc) in this case.
${\mathbf{m}}=0$
Indicates Cartesian coordinates.
${\mathbf{m}}=1$
Indicates cylindrical polar coordinates.
${\mathbf{m}}=2$
Indicates spherical polar coordinates.
Constraints:
• $0\le {\mathbf{m}}\le 2$ following a finite difference function;
• ${\mathbf{m}}=0$ following a Keller box scheme function.
3:     u[${\mathbf{npde}}×{\mathbf{npts}}$]const doubleInput
On entry: the PDE part of the original solution returned in the argument u by the PDE function.
Constraint: ${\mathbf{npde}}\ge 1$.
4:     nptsIntegerInput
On entry: the number of mesh points.
Constraint: ${\mathbf{npts}}\ge 3$.
5:     x[npts]const doubleInput
On entry: ${\mathbf{x}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$, must contain the mesh points as used by the PDE function.
6:     xp[intpts]const doubleInput
On entry: ${\mathbf{xp}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{intpts}}$, must contain the spatial interpolation points.
Constraint: ${\mathbf{x}}\left[0\right]\le {\mathbf{xp}}\left[0\right]<{\mathbf{xp}}\left[1\right]<\cdots <{\mathbf{xp}}\left[{\mathbf{intpts}}-1\right]\le {\mathbf{x}}\left[{\mathbf{npts}}-1\right]$.
7:     intptsIntegerInput
On entry: the number of interpolation points.
Constraint: ${\mathbf{intpts}}\ge 1$.
8:     itypeIntegerInput
On entry: specifies the interpolation to be performed.
${\mathbf{itype}}=1$
The solutions at the interpolation points are computed.
${\mathbf{itype}}=2$
Both the solutions and their first derivatives at the interpolation points are computed.
Constraint: ${\mathbf{itype}}=1$ or $2$.
9:     up[${\mathbf{npde}}×{\mathbf{intpts}}×{\mathbf{itype}}$]doubleOutput
Note: the element ${\mathbf{UP}}\left(i,j,k\right)$ is stored in the array element ${\mathbf{up}}\left[\left(k-1\right)×{\mathbf{npde}}×{\mathbf{intpts}}+\left(j-1\right)×{\mathbf{intpts}}+i-1\right]$.
On exit: if ${\mathbf{itype}}=1$, ${\mathbf{UP}}\left(\mathit{i},\mathit{j},1\right)$, contains the value of the solution ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{\mathrm{out}}\right)$, at the interpolation points ${x}_{\mathit{j}}={\mathbf{xp}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{intpts}}$ and $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
If ${\mathbf{itype}}=2$, ${\mathbf{UP}}\left(\mathit{i},\mathit{j},1\right)$ contains ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{\mathrm{out}}\right)$ and ${\mathbf{UP}}\left(\mathit{i},\mathit{j},2\right)$ contains $\frac{\partial {U}_{\mathit{i}}}{\partial x}$ at these points.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EXTRAPOLATION
On entry, interpolating point $〈\mathit{\text{value}}〉$ with the value $〈\mathit{\text{value}}〉$ is outside the x range.
NE_INT
On entry, ${\mathbf{intpts}}\le 0$: ${\mathbf{intpts}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{itype}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itype}}=1$ or $2$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}=0$, $1$ or $2$.
On entry, ${\mathbf{npde}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npde}}>0$.
On entry, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npts}}>2$.
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.
NE_NOT_STRICTLY_INCREASING
On entry, interpolation points xp badly ordered: $i=〈\mathit{\text{value}}〉$, ${\mathbf{xp}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{xp}}\left[j-1\right]=〈\mathit{\text{value}}〉$.
On entry, mesh points x badly ordered: $i=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[j-1\right]=〈\mathit{\text{value}}〉$.

## 7  Accuracy

See the PDE function documents.