Dissipative Numerical Methods for the Hunter-Saxton Equation

Abstract

In this paper, we present further development of the local discontinuous Galerkin (LDG) method designed in [21] and a new dissipative discontinuous Galerkin (DG) method for the Hunter-Saxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].