A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect

Authors:
Hailiang Liu;Jue Yan
Affiliations:
Iowa State University, Mathematics Department, Carver Hall 400, Ames, IA 50011, United States;Department of Mathematics, UCLA, Los Angeles, CA 90095, United States
Venue:
Journal of Computational Physics
Year:
2006

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Abstract

A local discontinuous Galerkin method for solving Korteweg-de Vries (KdV)-type equations with non-homogeneous boundary effect is developed. We provide a criterion for imposing appropriate boundary conditions for general KdV-type equations. The discussion is then focused on the KdV equation posed on the negative half-plane, which arises in the modeling of transition dynamics in the plasma sheath formation [H. Liu, M. Slemrod, KdV dynamics in the plasma-sheath transition, Appl. Math. Lett. 17(4) (2004) 401-410]. The guiding principle for selecting inter-cell fluxes and boundary fluxes is to ensure the L^2 stability and to incorporate given boundary conditions. The local discontinuous Galerkin method thus constructed is shown to be stable and efficient. Numerical examples are given to confirm the theoretical result and the capability of this method for capturing soliton wave phenomena and various boundary wave patterns.