Third order maximum-principle-satisfying and positivity-preserving Lax-Wendroff discontinuous Galerkin methods for hyperbolic conservation laws

Abstract

There have been intensive studies on maximum-principle-satisfying and positivity-preserving methods for hyperbolic conservation laws. Most of them are based on the method of lines type time marching approaches, e.g. the Runge-Kutta methods, multi-step methods and backward Euler method. As an alternative, the Lax-Wendroff time marching approach utilizes the information of PDEs in the Taylor expansion of the solution in time, hence it is a high order and single-stage method. In this work, we propose third order maximum-principle-satisfying and positivity-preserving schemes for scalar conservation laws and the Euler equations based on the Lax-Wendroff time discretization and discontinuous Galerkin spatial discretization. The accuracy and effectiveness of the maximum-principle-satisfying and positivity-preserving techniques are demonstrated by ample numerical tests.