Third M.I.T. Conference on Computational Fluid and Solid Mechanics June 14–17, 2005  

The heterogeneous multiscale method based on the discontinuous Galerkin and the finite volume methods for hyperbolic problems

Shanqin Chena, Weinan Eb, Chi-Wang Shua,*
aDivision of Applied Mathematics, Brown University, Providence, RI 02912, USA  bDepartment of Mathematics, Princeton University, Princeton, NJ 08544, USA
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ABSTRACT
In this paper, we review a discontinuous Galerkin (DG) method and develop a finite volume (FV) method, within the framework of the heterogeneous multiscale method (HMM), for solving hyperbolic problems. Although the methods can be applied to general cases, we consider in this paper only hyperbolic scalar advection equations and Euler systems. Error estimates are given for the linear equations and numerical results are provided for the linear and nonlinear problems to demonstrate the capability of the methods.

Keywords:  Heterogeneous multiscale method; Homogenization; Discontinuous Galerkin method; Finite volume method; Advection equation; Euler equations
* Corresponding author. Tel.: +1 401 863 2549; Fax: +1 401 863 1355; E-mail: shu@dam.brown.edu