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

• Published in: Journal of computational physics 2022-12, Vol.470, p.111591, Article 111591
• Main Authors: Xu, Ziyao, Shu, Chi-Wang
• Format: Article
• Language: English
• Subjects: Euler equations; Lax-Wendroff discontinuous Galerkin methods (LWDG); Maximum-principle-satisfying; Positivity-preserving; Scalar conservation laws
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111591

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. •Third order positivity-preserving Lax-Wendroff DG schemes for conservation laws are designed.•Lax-Wendroff procedure without certain mixed derivatives and suitable usage of DDG techniques ensure cell-average positivity.•The Zhang-Shu framework is then adopted with a scaling limiter which maintains high order accuracy when enforcing positivity.