A simple MATLAB program to compute differentiation matrices for arbitrary meshes via Lagrangian interpolation

TL;DR

This paper introduces a MATLAB tool for computing differentiation matrices on arbitrary 1D meshes using Lagrangian interpolation, unifying spectral and finite difference methods.

Contribution

It presents a versatile MATLAB program that computes differentiation matrices for arbitrary meshes, bridging spectral and finite difference approaches.

Findings

The program produces spectral differentiation matrices for Chebyshev or Legendre points.

For evenly spaced meshes with M<N and M odd, it matches central finite difference matrices.

The method generalizes differentiation matrix computation to arbitrary 1D meshes.

Abstract

A MATLAB program for computing differentiation matrices for arbitrary one-dimensional meshes is presented in this manuscript. The differentiation matrices for a mesh of N arbitrarily spaced points are formed from those obtained using Lagrangian interpolation on stencils of a fixed but arbitrary number M<=N of contiguous mesh points. For the particular case M=N and meshes with Chebyshev or Legendre distributions of points, the program yields the well known spectral differentiation matrices. For M<N and M odd, the differentiation matrices coincide, for the special case of an evenly spaced mesh, with those obtained by central finite differences.