Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system
Blanca Ayuso José A. Carrillo Chi-Wang Shu
Kinetic & Related Models 2011, 4(4): 955-989 doi: 10.3934/krm.2011.4.955
We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for all the proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.
keywords: discontinuous Galerkin energy conservation. mixed-finite elements Vlasov-Poisson system
Efficient time discretization for local discontinuous Galerkin methods
Yinhua Xia Yan Xu Chi-Wang Shu
Discrete & Continuous Dynamical Systems - B 2007, 8(3): 677-693 doi: 10.3934/dcdsb.2007.8.677
In this paper, we explore three efficient time discretization techniques for the local discontinuous Galerkin (LDG) methods to solve partial differential equations (PDEs) with higher order spatial derivatives. The main difficulty is the stiffness of the LDG spatial discretization operator, which would require a unreasonably small time step for an explicit local time stepping method. We focus our discussion on the semi-implicit spectral deferred correction (SDC) method, and study its stability and accuracy when coupled with the LDG spatial discretization. We also discuss two other time discretization techniques, namely the additive Runge-Kutta (ARK) method and the exponential time differencing (ETD) method, coupled with the LDG spatial discretization. A comparison is made among these three time discretization techniques, to conclude that all three methods are efficient when coupled with the LDG spatial discretization for solving PDEs containing higher order spatial derivatives. In particular, the SDC method has the advantage of easy implementation for arbitrary order of accuracy, and the ARK method has the smallest CPU cost in our implementation.
keywords: local discontinuous Galerkin method additive Runge-Kutta method higher order spatial derivatives. Spectral deferred correction method exponential time differencing method
Energy conserving local discontinuous Galerkin methods for wave propagation problems
Yulong Xing Ching-Shan Chou Chi-Wang Shu
Inverse Problems & Imaging 2013, 7(3): 967-986 doi: 10.3934/ipi.2013.7.967
Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.
keywords: superconvergence. local discontinuous Galerkin method Wave propagation error estimate energy conservation

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