An arbitrary high order series-based time-marching method with discontinuous Galerkin method for 2D convection problem

Abstract

This paper presents an arbitrary high-order series-based time-marching (SBTM) method to match with the high-order discontinuous Galerkin (DG) method in spatial discretization. And the detail of this hybrid method, SBTM-DG, is presented by solving a 2D convection problem where the exact solution of the semi-discrete system derived by DG method can be obtained by SBTM method. The SBTM-DG method is explicit single-step scheme of arbitrary high order of accuracy in time and with all advantages of DG method in space. The high order of accuracy in temporal direction is obtained by multiplications between small-scale matrixes and vectors in each cells with dimension K=(k+1)(k+2)/2, where k is the degree of basic function of DG method, which makes the SBTM-DG method memory-saving and efficient. With the help of projection, the error estimation between the exact solution and the full-discrete solution is O(h)k+1+O(Δt)S, where S is the degree of polynomial of time variable. By energy method we can prove the stability. By matrix method, we obtain the stability condition where time step Δt can be larger than mesh size h. Numerical examples confirm the analysis and show the good performance of SBTM-DG method. Furthermore, the convergence orders are around 20 about k, which makes SBTM-DG method significant.