Local Discontinuous Galerkin Methods to a Dispersive System of KdV-Type Equations

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the dispersive system of Korteweg–de Vries (KdV) type equations. Based on a cardinal conservative quantity of this system, we design and discuss two different types of numerical fluxes, including the conservative and dissipative ones for the linear and nonlinear terms respectively. Thus, one conservative together with three dissipative LDG schemes for the KdV-type system are developed in our paper. The invariant preserving property for the conservative scheme and corresponding dissipative properties for the other three dissipative schemes are all presented and proven in this paper. The error estimates for two schemes are given, whose numerical fluxes for linear terms are chosen as the dissipative type. Assuming that the discontinuous piecewise polynomials of degree less than or equal to k are adopted, and conservative numerical fluxes are employed to discretize the nonlinear terms, we obtain a suboptimal a priori bound of order k; yet in the case of dissipative fluxes, we obtain a slightly better bound of order \(k+\frac{1}{2}\). Numerical experiments for this system in different circumstances are provided, including accuracy tests for two kinds of traveling waves, long-time simulations for solitary waves and interactions of multi-solitary waves, to illustrate the accuracy and capability of these schemes.