# Pseudospectral Methods in NDSolve

What possibilites does NDSolve provide for solving partial differential equation using a pseudospectral method?

NDSolve provides the Option "DifferenceOrder"->"Pseudospectral". As I understand the documentary the method is using some form of Chebyshev polynomials $T_n$ for the expansion of a function $f$ to solve the differential equation:

$f(x)=\sum_{n=0}^{N}a_n T_n(x)$

• Am I correct that NDSolve really uses a pseudospectral method here or is it just distribution the gridpoints in a different way and using FD?
• If so, which polynomials are used here exactly? How is the truncation $N$ defined?
• Which possibilities do I have to implement a pseudospectral method inherently in Mathematica's NDSolve (if any) using standard polynomials (like Bessel Functions, Jacobi Polynomials etc)?
• My understanding is that it uses a Chebyshev grid and Chebyshev differentiation matrices to approximate spatial derivatives; this is equivalent, I believe, to a Chebyshev interpolation model that approximates the Chebyshev series (which approximation may be the pseudo- in pseudospectral). The spatial grid (N+1 points) determines the degree N. -- You can write your own solvers and plug them into NDSolve. – Michael E2 Jan 9 '18 at 18:40

• Oh Hello Rob! Good to see you here! NDSolve rocks! – user21 Dec 19 '18 at 18:41