naginterfaces.library.pde.dim1_​parab_​fd_​interp

naginterfaces.library.pde.dim1_parab_fd_interp(m, u, x, xp, itype)

dim1_parab_fd_interp 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 dim1_parab_fd(), dim1_parab_dae_fd() or dim1_parab_remesh_fd(), or one of the Keller box schemes dim1_parab_keller(), dim1_parab_dae_keller() or dim1_parab_remesh_keller().

For full information please refer to the NAG Library document for d03pz

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d03/d03pzf.html

Parameters

mint

The coordinate system used. If the call to dim1_parab_fd_interp 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 dim1_parab_fd_interp in this case.

$m=0$

Indicates Cartesian coordinates.

$m=1$

Indicates cylindrical polar coordinates.

$m=2$

Indicates spherical polar coordinates.

ufloat, array-like, shape $(\text{npde},\text{npts})$

The PDE part of the original solution returned in the argument $u$ by the PDE function.

xfloat, array-like, shape $\left(\text{npts}\right)$

$x[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{npts}$, must contain the mesh points as used by the PDE function.

xpfloat, array-like, shape $\left(\text{intpts}\right)$

$xp[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{intpts}$, must contain the spatial interpolation points.

itypeint

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.

Returns

upfloat, ndarray, shape $(\text{npde},\text{intpts},itype)$

If $itype=1$, $up[\text{i}-1,\text{j}-1,0]$, contains the value of the solution $U_{\text{i}}(x_{\text{j}},t_{out})$, at the interpolation points $x_{\text{j}}=xp[\text{j}-1]$, for $\text{i}=1,2,\dots ,\text{npde}$, for $\text{j}=1,2,\dots ,\text{intpts}$.

If $itype=2$, $up[\text{i}-1,\text{j}-1,0]$ contains $U_{\text{i}}(x_{\text{j}},t_{out})$ and $up[\text{i}-1,\text{j}-1,1]$ contains $\frac{\partial U_{\text{i}}}{\partial x}$ at these points.

Raises

NagValueError

(errno $1$)

On entry, $itype=⟨value⟩$.

Constraint: $itype=1$ or $2$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m=0$, $1$ or $2$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$, $x[\text{i}-1]=⟨value⟩$, $\text{j}=⟨value⟩$ and $x[\text{j}-1]=⟨value⟩$.

Constraint: $x\left[0\right]<x\left[1\right]<\cdots <x[\text{npts}-1]$.

(errno $1$)

On entry, $\text{intpts}=⟨value⟩$.

Constraint: $\text{intpts}\ge 1$.

(errno $1$)

On entry, $\text{npts}=⟨value⟩$.

Constraint: $\text{npts}>2$.

(errno $1$)

On entry, $\text{npde}=⟨value⟩$.

Constraint: $\text{npde}>0$.

(errno $2$)

On entry, $\text{i}=⟨value⟩$, $xp[\text{i}-1]=⟨value⟩$, $\text{j}=⟨value⟩$ and $xp[\text{j}-1]=⟨value⟩$.

Constraint: $x\left[0\right]\le xp\left[0\right]<xp\left[1\right]<\cdots <xp[\text{intpts}-1]\le x[\text{npts}-1]$.

(errno $3$)

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

Notes

dim1_parab_fd_interp 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. dim1_parab_fd_interp 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.