Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods

TL;DR

This paper investigates aliasing-induced instabilities in high-order discontinuous Galerkin methods for advection equations, comparing bounds and stabilisation techniques, and demonstrating the effectiveness of split form approaches with Gauss-Lobatto points.

Contribution

It provides a detailed analysis of aliasing errors, compares energy bounds, and shows that split form stabilisation effectively prevents aliasing in DG schemes with SBP-SAT properties.

Findings

Elliptic norm bounds better predict PDE behaviour than $L^2$ bounds.

Over-integration does not generally reduce aliasing errors.

Split form methods ensure aliasing-free solutions in compatible DG schemes.

Abstract

We analyse instabilities due to aliasing errors when solving one dimensional non-constant advection speed equations and discuss means to alleviate these types of errors when using high order discontinuous Galerkin (DG) schemes. First, we compare analytical bounds for the continuous and discrete version of the PDEs. Whilst traditional $L^2$ norm energy bounds applied to the discrete PDE do not always predict the physical behaviour of the continuous version of the equation, more strict elliptic norm bounds correctly bound the behaviour of the continuous PDE. Having derived consistent bounds, we analyse the effectiveness of two stabilising techniques: over-integration and split form variations (conservative, non-conservative and skew-symmetric). Whilst the former is shown to not alleviate aliasing in general, the latter ensures an aliasing-free solution if the splitting form of the discrete PDE is consistent with the continuous equation. The success of split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term (SBP-SAT) properties (e.g. DG with Gauss-Lobatto points). Numerical experiments are included to illustrate the theoretical findings.