Commun. Comput. Phys., 7 (2010), pp. 1-46.

Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations

Yan Xu 1, Chi-Wang Shu 2*

1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

Received 29 January 2009; Accepted (in revised version) 22 May 2009
Available online 10 June 2009


Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, less restriction in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and local data structure and the resulting high parallel efficiency. In this paper, we give a general review of the local DG (LDG) methods for solving high-order time-dependent partial differential equations (PDEs). The important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, is highlighted. Some of the applications of the LDG methods for high-order time-dependent PDEs are also be discussed.

AMS subject classifications: 65M60

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Key words: Discontinuous Galerkin method, local discontinuous Galerkin method, numerical flux, stability, time discretization, high order accuracy, stability, error estimates.

*Corresponding author.
Email: (Y. Xu), (C.-W. Shu)

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