Local DG method using WENO type limiters for convection–diffusion problems

The local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection–diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as Runge–Kutta LDG (RKLDG) when TVD Runge–Kutta method is applied for time discretization. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper, we investigated RKLDG methods with WENO and Hermite WENO (HWENO) limiters for solving convection–diffusion equations on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition. Numerical results are provided to illustrate the behavior of these procedures.