Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients

TL;DR

This paper investigates the long-term error behavior of discontinuous Galerkin methods with variable coefficients for hyperbolic PDEs, deriving conditions for bounded errors and analyzing the impact of flux choices.

Contribution

It extends error boundedness analysis to variable coefficients in DG methods, providing conditions for bounded errors and insights into flux effects.

Findings

Bounded error conditions are derived for variable coefficient problems.

Upwind fluxes can maintain bounded errors, while central fluxes may lead to unbounded growth.

Numerical tests confirm theoretical predictions about error behavior.

Abstract

For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework. For linear problems with constant coefficients, it is well-known in the literature that the choice of the numerical flux (e.g. central or upwind) and the selection of the polynomial basis (e.g. Gau{\ss}-Legendre or Gau{\ss}-Lobatto-Legendre) affects both the growth rate and the asymptotic value of the error. Here, we extend these investigations of the long time error to variable coefficients using both Gau{\ss}-Lobatto-Legendre and Gau{\ss}-Legendre nodes as well as several numerical fluxes. We derive conditions guaranteeing that the errors are still bounded in time. Furthermore, we analyse the error behaviour under these conditions and demonstrate in several numerical tests similarities to the case of constant coefficients. However, if these conditions are violated, the error shows a completely different behaviour. Indeed, by applying central numerical fluxes, the error increases without upper bound while upwind numerical fluxes can still result in uniformly bounded numerical errors. An explanation for this phenomenon is given, confirming our analytical investigations.