Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes

Abstract

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection–diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws (Perthame, 1996) and (Zhang, 2010) [14,26], we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection–diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in Zhang (2012) [30], we extend the schemes to two-dimensional convection–diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier–Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.