Spectral shock detection for dynamically developing discontinuities

TL;DR

This paper enhances spectral shock detection methods to dynamically identify true discontinuities in solutions of PDEs, improving accuracy in simulations involving evolving shocks and reducing spurious oscillations.

Contribution

It introduces improved techniques for detecting and handling developing discontinuities in spectral methods, addressing challenges of dynamic shock formation and oscillation suppression.

Findings

Successfully detects true shocks in Burgers' equation

Reduces spurious oscillations in spectral solutions

Automatically identifies cases needing post-processing

Abstract

Pseudospectral schemes are a class of numerical methods capable of solving smooth problems with high accuracy thanks to their exponential convergence to the true solution. When applied to discontinuous problems, such as fluid shocks and material interfaces, due to the Gibbs phenomenon, pseudospectral solutions lose their superb convergence and suffer from spurious oscillations across the entire computational domain. Luckily, there exist theoretical remedies for these issues which have been successfully tested in practice for cases of well defined discontinuities. We focus on one piece of this procedure---detecting a discontinuity in spectral data. We show that realistic applications require treatment of discontinuities dynamically developing in time and that it poses challenges associated with shock detection. More precisely, smoothly steepening gradients in the solution spawn spurious oscillations due to insufficient resolution, causing premature shock identification and information loss. We improve existing spectral shock detection techniques to allow us to automatically detect true discontinuities and identify cases for which post-processing is required to suppress spurious oscillations resulting from the loss of resolution. We then apply these techniques to solve an inviscid Burgers' equation in 1D, demonstrating that our method correctly treats genuine shocks caused by wave breaking and removes oscillations caused by numerical constraints.