Invited Lecture
Explicit-implicit-null (EIN) time-marching for high order PDEs
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Abstract
Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k is at least 2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction $\tau =O(h^k)$ (where $\tau$ and $ h $ are the time step size and spatial mesh size respectively) 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 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 high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations. For simplified equations with constant coefficients, we perform analysis to guide the choice of the coefficient for the added and subtracted highest order derivative terms in order to guarantee stability for large time steps. 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.
This talk is based on joint work with Haijin Wang, Qiang Zhang and Shiping Wang, and with Meiqi Tan and Juan Cheng.