High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms

Journal Article · Sat May 20 04:00:00 EDT 2006 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2005.10.005· OSTI ID:20767046

Yulong, Xing ^[1]; Chiwang, Shu ^[2]

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1. Department of Mathematics, Brown University, Providence, RI 02912 (United States)
2. Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912 (United States)

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206-227; J. Sci. Comput., accepted], we designed a well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, 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.

• OSTI ID: 20767046
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 2 Vol. 214; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ACCURACY
BALANCES
CONSERVATION LAWS
FINITE ELEMENT METHOD
GALERKIN-PETROV METHOD
GEOMETRY
NOZZLES
SIMULATION
SOURCE TERMS
STEADY-STATE CONDITIONS
TWO-DIMENSIONAL CALCULATIONS
TWO-PHASE FLOW
WATER
WAVE EQUATIONS