Discrete conservation laws and the convergence of long time simulations of the mkdv equation

Authors:
C. Gorria;M. A. Alejo;L. Vega
Affiliations:
Department of Applied Mathematics and Statistics, The University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain;Department of Mathematics, The University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain;Department of Mathematics, The University of the Basque Country, P.O. Box 644, 48080 Bilbao, Spain
Venue:
Journal of Computational Physics
Year:
2013

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Abstract

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.