Interpolation

Topics
	Interpolation at grid elements
	Interpolation schemes for grids elements (centers, faces, vertices)
	Interpolation at points
	Module for point (anywhere) interpolation.

Modules
module	mod_interpolation_init
	Initialization of declared interpolation schemes.

Detailed Description

Interpolation is defined as the way to recover a continuous function based on discrete node values. Most of the interpolation schemes present here are based on Lagrange interpolation.

It is worth to note that Lagrange interpolation is, mathematicaly, the same thing as the reconstruction of a polynomial based on known discrete derivatives.

Example

The fourth order Lagrange interpolating polynomial is:

\[ P^4(x) = \sum_{j=0}^{3} P_j(x) P_j(x) = \phi_j \prod_{k=0,k \neq j}^{3} \frac{x-x_k}{x_j - x_k} \]

which is algebricaly equal to the \(4^{th}\) order discrete polynomial \(Q^4(x)\) based on discrete derivatives:

\[ Q^4(x) = \phi_1 + x D'_{3}(x) + \frac{x^2}{2} D''_{2}(x) + \frac{x^3}{6} D'''_{1}(x) \]

where \( D'_{3} \) is the \(3^{rd}\) order forward discrete first derivative based on the \(\phi\) values.

Note

Extrapolation is based on the same principle as interpolation. However, the term is preferably used when discussing about interpolating outside the stencil. For instance, a fourth order extrapolation is mathematicaly the same thing as a third order interpolation for any given \(x\).