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

naginterfaces.library.pde.dim1_parab_coll_interp(u, xbkpts, npoly, xp, itype, comm)

dim1_parab_coll_interp may be used in conjunction with either dim1_parab_coll() or dim1_parab_dae_coll(). It computes the solution and its first derivative at user-specified points in the spatial coordinate.

For full information please refer to the NAG Library document for d03py

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d03/d03pyf.html

Parameters

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

The PDE part of the original solution returned in the argument $u$ by the function dim1_parab_coll() or dim1_parab_dae_coll().

xbkptsfloat, array-like, shape $\left(\text{nbkpts}\right)$

$xbkpts[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{nbkpts}$, must contain the break-points as used by dim1_parab_coll() or dim1_parab_dae_coll().

npolyint

The degree of the Chebyshev polynomial used for approximation as used by dim1_parab_coll() or dim1_parab_dae_coll().

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 solution at the interpolation points are computed.

$itype=2$

Both the solution and the first derivative at the interpolation points are computed.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by prior calls to dim1_parab_coll() or dim1_parab_dae_coll().

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

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

(errno $1$)

On entry, $\text{npts}=⟨value⟩$, $\text{nbkpts}=⟨value⟩$ and $npoly=⟨value⟩$.

Constraint: $\text{npts}=(\text{nbkpts}-1)\times npoly+1$.

(errno $1$)

On entry, $\text{intpts}=⟨value⟩$.

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

(errno $1$)

On entry, $npoly=⟨value⟩$.

Constraint: $npoly>0$.

(errno $1$)

On entry, $\text{nbkpts}=⟨value⟩$.

Constraint: $\text{nbkpts}\ge 2$.

(errno $1$)

On entry, $\text{npde}=⟨value⟩$.

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

(errno $2$)

On entry, $itype=2$ and at least one interpolation point coincides with a break-point, i.e., interpolation point no $⟨value⟩$ with value $⟨value⟩$ is close to break-point $⟨value⟩$ with value $⟨value⟩$.

(errno $2$)

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

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

(errno $3$)

Extrapolation is not allowed.

Notes

dim1_parab_coll_interp is an interpolation function for evaluating the solution of a system of partial differential equations (PDEs), or the PDE components of a system of PDEs with coupled ordinary differential equations (ODEs), at a set of user-specified points. The solution of a system of equations can be computed using dim1_parab_coll() or dim1_parab_dae_coll() on a set of mesh points; dim1_parab_coll_interp can then be employed to compute the solution at a set of points other than those originally used in dim1_parab_coll() or dim1_parab_dae_coll(). It can also evaluate the first derivative of the solution. Polynomial interpolation is used between each of the break-points $xbkpts[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{nbkpts}$. When the derivative is needed ($itype=2$), the array $xp[\text{intpts}-1]$ must not contain any of the break-points, as the method, and consequently the interpolation scheme, assumes that only the solution is continuous at these points.