Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

Abstract

Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k≥2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ=O(hk) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method discussed in this paper is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss the high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations, respectively. By the aid of the Fourier method, we perform stability analysis for the schemes on the simplified equations with periodic boundary conditions, which demonstrates the stability criteria for the resulting schemes. Even though the analysis is only performed on the simplified equations, numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations.