Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations
Journal of Computational Physics
hp-Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization
Journal of Scientific Computing
A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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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).