Insights on Aliasing Driven Instabilities for Advection Equations with Application to Gauss–Lobatto Discontinuous Galerkin Methods

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.

A free–energy stable nodal discontinuous Galerkin approximation with summation–by–parts property for the Cahn–Hilliard equation

“…Although it is less accurate than its Gauss counterpart (for the same number of quadrature nodes), it satisfies the SBP-SAT property, thus allowing one to construct schemes that are provably stable [4]. Precisely, different authors have presented energy-and entropy-stable schemes using this framework for the linear advection equation [5,6], Burgers equation [7,8], shallow water equations [9], Euler and Navier-Stokes equations [10,8], and the magneto-hydrodynamics equations [11], among others.…”

On conservation and stability properties for summation-by-parts schemes

“…In this section, a general notion of summation-by-parts (SBP) operators for semidiscretisations of conservation laws will be presented using the notation of [69,70]. Tables containing translations to the notations used in the finite difference community [59] and the finite element framework [47] can be found in A.…”

“…The split and unsplit fluxes use possibility ii) with two different choices of the interpolation. In order to help the reader avoid misunderstandings on the notation, some translation rules to the finite difference setting of [59] and the discontinuous Galerkin spectral element framework of [38,47] are presented in A.…”

“…However, if generalised SBP operators are used, several problems have to be handled for non-diagonal norms [75] and nodal bases not including boundary nodes, as can be seen in recent investigations by Nordström and Ruggiu [59] and Manzanero, Rubio, Ferrer, Valero, and Kopriva [47]. In order to remedy these problems, corrections for generalised SBP operators will be presented in this article, extending the work of [67,69].…”

“…Introducing new corrections, generalised SBP operators are proven to be both conservative and stable. Similarly, the weighted L 2 framework of Manzanero, Rubio, Ferrer, Valero, and Kopriva [47] is applied in section 4, yielding new conservative and stable discretisations using general SBP operators. Thereafter, numerical results for the new methods are presented in section 5, including convergence studies and eigenvalue analyses.…”