The discontinuous Galerkin method for general nonlinear third-order ordinary differential equations

Abstract

In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L2-stable and provides the optimal (p+1)-th order of accuracy for smooth solutions when using piecewise p-th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order p+1 toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Several numerical results are provided to confirm the convergence of the proposed scheme.