Spectral Methods in the Presence of Discontinuities
For researchers using spectral methods to solve differential equations with discontinuities, this work provides a method to regain exponential convergence, which is crucial for accurate simulations of realistic systems.
The paper addresses the poor convergence of spectral methods for non-smooth problems and improves a post-processing technique (optimally convergent mollifiers) to recover exponential convergence from spectral reconstructions of discontinuous data.
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.