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Structural Stability of Discontinuous Galerkin Schemes

Structural Stability of Discontinuous Galerkin Schemes The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193–2199, 2009; Chaos Solitons Fractals 41(2):655–660, 2009). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Structural Stability of Discontinuous Galerkin Schemes

Acta Applicandae Mathematicae , Volume 113 (1) – Aug 18, 2010

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References (31)

Publisher
Springer Journals
Copyright
Copyright © 2010 by Springer Science+Business Media B.V.
Subject
Mathematics; Mechanics; Statistical Physics, Dynamical Systems and Complexity; Theoretical, Mathematical and Computational Physics; Computer Science, general; Mathematics, general
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-010-9583-5
Publisher site
See Article on Publisher Site

Abstract

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193–2199, 2009; Chaos Solitons Fractals 41(2):655–660, 2009).

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Aug 18, 2010

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