×

Jacobian-free incomplete Riemann solvers. (English) Zbl 1407.65142

Klingenberg, Christian (ed.) et al., Theory, numerics and applications of hyperbolic problems I, Aachen, Germany, August 2016. Cham: Springer. Springer Proc. Math. Stat. 236, 295-307 (2018).
Summary: The purpose of this work is to present some recent developments about incomplete Riemann solvers for general hyperbolic systems. Polynomial viscosity matrix (PVM) methods based on internal approximations to the absolute value function are introduced, and they are compared with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Moreover, they can be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions. Some numerical experiments involving the relativistic magnetohydrodynamic equations are presented, both in one and two dimensions. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate, running stable under a satisfactory time step restriction.
For the entire collection see [Zbl 1398.65011].

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
35L02 First-order hyperbolic equations
35L65 Hyperbolic conservation laws
76M12 Finite volume methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI