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

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 L2 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.