Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation (Communications in Computational Physics, 2020)
In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in onedimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do notneed numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L2 error estimates are establishedfor the linear and nonlinear equation using several projections for different parameterchoices. 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 usingpiecewise k-th degree polynomials for many cases.
Keywords: Korteweg-de Vries equation; central DG method; stability; error estimates.