A discontinuous Galerkin method and its error estimate for nonlinear fourth-order wave equations

Abstract

In this paper, an ultra-weak local discontinuous Galerkin (UWLDG) method for a class of nonlinear fourth-order wave equations is designed and analyzed. The UWLDG method is a new DG method designed for solving partial differential equations (PDEs) with high order spatial derivatives. We prove the energy conserving property of our scheme and its optimal error estimates in the L2-norm for the solution itself as well as for the auxiliary variables approximating the derivatives of the solution. Compatible high order energy conserving time integrators are also proposed. The theoretical results are confirmed by numerical experiments.