Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs

¹ College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu Province, P.R. China.
hjwang@njupt.edu.cn
² Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China .
qzh@nju.edu.cn
³ Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
shu@dam.brown.edu

Received: 29 December 2016
Revised: 9 March 2017
Accepted: 2 April 2017

Abstract

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.