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

• Published in: Journal of computational physics Vol. 214; no. 2; pp. 567 - 598
• Main Authors: Xing, Yulong, Shu, Chi-Wang
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 20.05.2006; Elsevier
• Subjects: ACCURACY; BALANCES; Chemosensitive movement; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational techniques; Computer simulation; CONSERVATION LAWS; Discontinuous Galerkin method; Elastic wave equation; Exact sciences and technology; FINITE ELEMENT METHOD; Galerkin methods; GALERKIN-PETROV METHOD; GEOMETRY; High order accuracy; Hyperbolic balance laws; Hyperbolic systems; Laws; Mathematical analysis; Mathematical methods in physics; Mathematical models; Nozzle flow; NOZZLES; Physics; Runge-Kutta method; Shallow water equation; SIMULATION; Source term; SOURCE TERMS; STEADY-STATE CONDITIONS; TWO-DIMENSIONAL CALCULATIONS; TWO-PHASE FLOW; WATER; WAVE EQUATIONS; WENO scheme; Conservation laws; Discontinuous Galerkin method; Two phase flow; Shallow water equation; Elastic wave equation; Nozzle flow; WENO scheme; High order accuracy; Source term; Chemosensitive movement; Hyperbolic balance laws; Discontinuity; Difference scheme; Hyperbolic system; Wave equations; Hyperbolic equations; Two dimensional model; Calculation methods; Galerkin-Petrov method; Finite element method; Two-phase flow; Calculation; Steady state solution; Shallow-water equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2005.10.005

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.