Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments
In these lectures we give a general survey on discontinuous Galerkin methods for solving time-dependent partial differential equations. We also present a few recent developments on the design, analysis, and application of these discontinuous Galerkin meth
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troduction. Discontinuous Galerkin (DG) methods belong to the class of finite element methods. The finite element function space corresponding to DG methods consists of piecewise polynomials (or other simple functions) which are allowed to be completely discontinuous across element interfaces. Therefore, using finite element terminologies, DG methods are the most extreme case of nonconforming finite element methods. The first DG method was introduced in 1973 by Reed and Hill in a Los Alamos technical report [72]. It solves the equations for neutron transport, which are time independent linear hyperbolic equations. A major development of the DG method is carried out by Cockburn et al. in a series of papers [23, 25, 27–29], in which the authors have established a framework to easily solve nonlinear time-dependent hyperbolic equations, such as the Euler equations of compressible gas dynamics. The DG method of Cockburn et al. belongs to the class of method-of-lines, namely the DG discretization is used only for the spatial variables, and explicit, nonlinearly stable high order Runge–Kutta methods [81] are used to discretize the time variable. Other important features of the DG method of Cockburn et al. include the usage of exact or approximate Riemann solvers as interface fluxes and total variation bounded (TVB) nonlinear limiters [79] to achieve non-oscillatory properties for strong shocks, both of which are borrowed from the methodology of high resolution finite volume schemes. The DG method has found rapid applications in such diverse areas as aeroacoustics, electro-magnetism, gas dynamics, granular flows, magnetohydrodynamics, meteorology, modeling of shallow water, oceanography, oil recovery simulation, semiconductor device simulation, transport of contaminant in porous media, turbomachinery, turbulent flows, viscoelastic ∗ Division of Applied Mathematics, Brown University, Providence, RI 02912, USA, [email protected]
X. Feng et al. (eds.), Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, The IMA Volumes in Mathematics and its Applications 157, DOI 10.1007/978-3-319-01818-8 2, © Springer International Publishing Switzerland 2014
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flows and weather forecasting, among many others. For earlier work on DG methods, we refer to the survey paper [24], and other papers in that Springer volume, which contains the conference proceedings of the First International Symposium on Discontinuous Galerkin Methods held at Newport, Rhode Island in 1999. The lecture notes [21] is a good reference for many details, as well as the extensive review paper [31]. The review paper [99] covers the local DG method for partial differential equations (PDEs) containing higher order spatial derivatives. More recently, there are three special journal issues devoted to the DG method [32, 33, 35], which contain many interesting papers on DG method in all aspects including algorithm design, analysis, implementation, and applications. There are also a few recent books and lecture notes
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