The Local Discontinuous Galerkin Method Coupled With IMEX Time Marching

In this paper,we propose a class of local discontinuous Galerkin?LDG?method coupled with implicit-explicit?IMEX?multistep methods for solving the nonlinear partial differential equations?NPDEs?.The equations we solved in the paper contain the nonlinear Cahn-Hilliard equation and Schr?dinger equation,which have high order spatial derivation.Thus we take the LDG method to rewrite the equations as the equivalent first order system.Given the constriction of CFL?Courant,Friedrichs,Lewy,three authors who propose the condition?condition,we adopt the IMEX multistep method to discretize temporal derivation,aiming to reducing the time step restriction from the fully explicit time marching.Via the schemes proposed,we could take larger time steps,from??⁴?to????Cahn-Hilliard equation?and from??²?to?1??Schr?dinger equation?.The numerical results in this paper validate that our IMEX LDG schemes are stable and efficient.The research of this paper is divided into two points as follows.Firstly,we propose the fully discrete IMEX LDG numerical scheme for the nonlinear Cahn-Hilliard equation.The nonlinearity of the diffusion mobility(7??adds the time step limitation and the difficulty to solve the nonlinear Cahn-Hilliard equation.Thus,before we implement the numerical scheme,we add and subtract a linear term(6₀?in the right hand side of the Cahn-Hilliard equation.This treatment reduce the proportion of the nonlinear term in the Cahn-Hilliard equation,which can reduce the difficulty of solution and the constriction of time step.The coefficients of the system are given in this paper.At the end of this part,a series of numerical experiments are given to validate the stability and efficiency of our IMEX LDG numerical scheme.Secondly,we construct the fully discrete IMEX LDG numerical scheme for the nonlinear Schr?dinger equation.The unknown function?in Schr?dinger equation is complex function.So,before we implement the numerical scheme,we decompose function?into its real and imaginary parts.Then we implement the IMEX LDG numerical scheme for the rewritten nonlinear system.The coefficients of the system are given in this part.At the end of the second part,we propose a series of numerical experiments.The results show that our scheme achieves the optimal convergence in^?norm.Meanwhile,the numerical results validate the stability and efficiency of the IMEX LDG scheme.