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Structural stability of finite dispersion-relation preserving schemes

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  • David, Claire
  • Sagaut, Pierre

Abstract

The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a 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 occurrance 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 domains. Such a behavior is referred here to has 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 solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes [1].

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  • David, Claire & Sagaut, Pierre, 2009. "Structural stability of finite dispersion-relation preserving schemes," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2193-2199.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:4:p:2193-2199
    DOI: 10.1016/j.chaos.2008.08.028
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    References listed on IDEAS

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    1. Feng, Zhaosheng & Chen, Goong, 2005. "Solitary wave solutions of the compound Burgers–Korteweg–de Vries equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 352(2), pages 419-435.
    2. David, Claire & Sagaut, Pierre, 2009. "Spurious solitons and structural stability of finite-difference schemes for non-linear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 655-660.
    3. David, Claire & Fernando, Rasika & Feng, Zhaosheng, 2007. "On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(1), pages 44-50.
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    1. David, Claire & Sagaut, Pierre, 2016. "Structural stability of Lattice Boltzmann schemes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 1-8.

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    1. David, Claire & Sagaut, Pierre, 2016. "Structural stability of Lattice Boltzmann schemes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 1-8.
    2. David, Claire & Sagaut, Pierre, 2009. "Spurious solitons and structural stability of finite-difference schemes for non-linear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 655-660.
    3. David, Claire & Fernando, Rasika & Feng, Zhaosheng, 2007. "On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(1), pages 44-50.

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