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

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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 the split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term properties (e.g. DG with Gauss–Lobatto points). Numerical experiments are included to illustrate the theoretical findings.

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Acknowledgements

The authors thank the European Commission for the financial support received through the project SSEMID: Stability and Sensitivity Methods for Industrial Design, under grant contract PITN-GA-675008. The authors acknowledge the computer resources and technical assistance provided by the Centro de Supercomputación y Visualización de Madrid (CeSViMa). This work was supported by a grant from the Simons Foundation (# 426393, David A. Kopriva).

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Authors and Affiliations

  1. ETSIAE-UPM - School of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, 28040, Madrid, Spain

    Juan Manzanero, Gonzalo Rubio, Esteban Ferrer & Eusebio Valero

  2. Department of Mathematics, The Florida State University, Tallahassee, FL, 32306, USA

    David A. Kopriva

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  1. Juan Manzanero
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  2. Gonzalo Rubio
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  3. Esteban Ferrer
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  4. Eusebio Valero
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  5. David A. Kopriva
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Correspondence to Juan Manzanero.

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Manzanero, J., Rubio, G., Ferrer, E. et al. Insights on Aliasing Driven Instabilities for Advection Equations with Application to Gauss–Lobatto Discontinuous Galerkin Methods. J Sci Comput 75, 1262–1281 (2018). https://doi.org/10.1007/s10915-017-0585-6

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  • DOI: https://doi.org/10.1007/s10915-017-0585-6

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