On maximum-principle-satisfying high order schemes for scalar conservation laws

By Xiangxiong Zhang, Chi-Wang Shu. Journal of Computational Physics, 2010

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On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes (2010, methodology): “Genuinely high order accurate finite volume and discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle have been constructed recently in [26].”
Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments (2014, background): “They can guarantee maximum principle in the scalar case [111, 115] and positivitypreserving for certain systems, for example positivity-preserving for density and pressure for Euler equations of compressible gas dynamics [112, 113, 115] and positivity-preserving for water height for shallow water eq”
A Survey of High Order Schemes for the Shallow Water Equations (2014, methodology): “Following the approaches in [43, 61, 65, 67], we have the following result.”