A local discontinuous Galerkin method for the Burgers–Poisson equation

In this work, we design, analyze and test a local discontinuous Galerkin method for solving the Burgers–Poisson equation. This model, proposed by Whitham [Linear and nonlinear waves, 1974] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method.