Spectral Methods in the Presence of Discontinuities

TL;DR

This paper investigates spectral methods for solving differential equations with discontinuities, focusing on improving convergence rates through mollifier-based post-processing techniques to handle non-smooth problems effectively.

Contribution

It introduces and analyzes optimally convergent mollifiers to enhance spectral method performance on non-smooth problems, achieving near-exponential convergence.

Findings

Mollifier-based post-processing improves convergence for discontinuous problems.

Spectral methods can be adapted for non-smooth data with the proposed techniques.

Enhanced convergence rates enable more accurate simulations of realistic systems.

Abstract

Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and $L_2$-norm. However, for non-smooth problems, convergence is significantly worse---the $L_2$-norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique---optimally convergent mollifiers---to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth data. This is an important first step towards using these techniques for simulations of realistic systems.