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Solvers for the high-order Riemann problem for hyperbolic balance laws. (English) Zbl 1148.65066
The authors study three methods for solving the Cauchy problem for a system of nonlinear 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 in the paper are new; one of the them results from a re-interpretation of the high-order numerical methods of A. Harten, B. Engquist, S. Osher, and S. Chakravarthy [J. Comput. Phys. 71, 231–303 (1987; Zbl 0652.65067)], and the other is a modification of the solver of E. F. Toro and V. A. Titarev [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2018, 271–281 (2002; Zbl 1019.35061)].
Some interesting results are presented. In particular, 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. The authors also address the question of balance between flux gradients and source terms, for steady flow. It is found that the arbitrary accuracy derivative (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.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76N15 Gas dynamics (general theory)
76M20 Finite difference methods applied to problems in fluid mechanics
Full Text: DOI
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