Study On Time Splitting Algorithms Of Local Discontinuous Galerkin Method For 2-Dimensional Diffusion-Reaction Problems

This paper mainly study the time discretization techniques of Local Discontinuous Galerkin (LDG) method on high dimensional space for the time-dependent diffusion-reaction problems. According to the characteristics of parabolic equations, implicit scheme of time-marching is a common method, which may require the solution of a large algebraic equations at each time level, and the computational efficient of LDG method reduce naturally. To this end, we use the operator splitting algorithms to develop the computational efficiency with a considerable saving in computing time and data memory. The algorithms include two parts:first, split multidimensional diffusion terms from reaction terms; second, consider the Local One Dimensional (LOD) technique while solving multidimensional diffusion terms. Splitting schemes we mainly discuss are Splitting-Backward Euler(SBE), Strang Splitting(symmetrical splitting), and Richardson extrapolation, whose time accuracy can be up to 1st,2nd and 4th order respectively. Besides, we also consider the boundary treatment of aperiodic boundary conditions (such as including Dirichlet and Neumann boundary condition) for the intermediate vectors in splitting, and by the boundary correction techniques the time accuracy reduction can be restored. Numerical experiments are presented to show that these splitting methods are very efficient solvers. Finally some application are also given, including the typical quenching problems, flame problems on 2D, and the Allen-Cahn equations.