High-order implicit maximum-principle-preserving local discontinuous Galerkin methods for convection–diffusion equations

We consider maximum-principle-preserving (MPP) property of two types of implicit local discontinuous Galerkin (LDG) schemes for solving diffusion and convection–diffusion equations. The first one is the original LDG scheme proposed in Cockburn and Shu (1998) with backward Euler time discretization. The second one adds an MPP scaling limiter defined in Zhang and Shu (2010), to the first one. Compared with explicit time discretization, implicit method allows for a larger time step. For pure diffusion equations in 1D, we prove that the second type of the LDG schemes is MPP, which can also achieve high order accuracy. This result can be generalized to 2D by using tensor product meshes but only for the second order $Q^1$ case. For convection–diffusion equations, the first type of LDG schemes, in the second order $P^1$ case in 1D, is proved to be MPP. In all the results above, in order to achieve the MPP property, it is necessary to have a lower bound on the time step in terms of the Courant–Friedrichs–Lewy (CFL) number. Although the analysis is only performed on linear equations, numerical experiments are provided to demonstrate that the second type of the LDG schemes works well in terms of the MPP property both for nonlinear convection–diffusion equations and for 2D higher order cases.