Solvers for the high-order Riemann problem for hyperbolic balance laws

• Published in: Journal of computational physics Vol. 227; no. 4; pp. 2481 - 2513
• Main Authors: Castro, C.E., Toro, E.F.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.02.2008; Elsevier
• Subjects: ADER methods; Classical Riemann problem; Computational techniques; Euler equations; Exact sciences and technology; Godunov method; High-order Riemann problem; Hyperbolic equations; Mathematical methods in physics; Physics; Shallow water equations; Source terms; Unstructured meshes; Well-balanced schemes; Shallow water equations; Unstructured meshes; Source terms; Well-balanced schemes; Hyperbolic equations; Classical Riemann problem; High-order Riemann problem; ADER methods; Godunov method; Euler equations; Discontinuity; Godunov scheme; Riemann problem; Numerical method; Finite volume methods; Calculation methods; Cauchy problem; Space-time; Advection; Initial conditions; Convergence rate; Calculation; Shallow-water equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2007.11.013

We study three methods for solving the Cauchy problem for a system of non-linear hyperbolic balance laws with initial condition consisting of two smooth vectors, with a discontinuity at the origin, a high-order Riemann problem. Two of the methods are new; one of the them results from a re-interpretation of the high-order numerical methods proposed by Harten et al. [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accuracy essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231–303] and the other is a modification of the solver in [E.F. Toro, V.A. Titarev, Solution of the generalised Riemann problem for advection-reaction equations, Proc. Roy. Soc. London A 458 (2002) 271–281]. A systematic assessment of all three solvers is carried out and their relative merits are discussed. We also implement the solvers, locally, in the context of high-order finite volume numerical methods of the ADER type, on unstructured meshes. Schemes of up to fifth order of accuracy in space and time for the two-dimensional compressible Euler equations and the shallow water equations with source terms are constructed. Empirically obtained convergence rates are studied systematically and, for the tests considered, these correspond to the theoretically expected orders of accuracy. We also address the question of balance between flux gradients and source terms, for steady flow. We find that the ADER schemes may be termed asymptotically well-balanced, in the sense that the well-balanced property is attained as the order of the method increases, and this without introducing any ad-hoc fixes to the schemes or the equations.