Commun. Comput. Phys., 1 (2006), pp. 100-134.

A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

Yulong Xing 1, Chi-Wang Shu 2*

1 Department of Mathematics, Brown University, Providence, RI 02912, USA.
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

Received 6 August 2005; Accepted (in revised version) 22 September 2005


Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work \cite{XS,XS2,XS3}, we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method. The same idea can be applied to the finite volume WENO schemes. We will first describe the algorithms and prove the well balanced property for the shallow water equations, and then show that the result can be generalized to a class of other balance laws. We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

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Key words: Hyperbolic balance laws; WENO finite volume scheme; discontinuous Galerkin method; high order accuracy; source term; conservation laws; shallow water equation; elastic wave equation; chemosensitive movement; nozzle flow; two phase flow.

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Correspondence to: Chi-Wang Shu , Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Email:
† Research supported by ARO grant W911NF-04-1-0291, NSF grant DMS-0510345 and AFOSR grant FA9550-05-1-0123.

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