Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L² error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.