Solvers for the HighOrder Riemann Problem for Hyperbolic Balance Laws
Castro, CristÃ³bal E., and Eleuterio F. Toro (2008),
Solvers for the HighOrder Riemann Problem for Hyperbolic Balance Laws,
J. Comput. Phys., 227(4), 24812513, doi:10.1016/j.jcp.2007.11.013.
 Abstract
 We 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 highorder Riemann problem. Two of the methods are new; one of the them results from a reinterpretation of the highorder numerical methods proposed by Harten et al. [17] and the other is a modification of the solver in [44]. 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 highorder 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 twodimensional 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 wellbalanced, in the sense that the wellbalanced property is attained as the order of the method increases, and this without introducing any adhoc fixes to the schemes or the equations.
 BibTeX

@article{id1011,
author = {Crist{\'o}bal E. Castro and Eleuterio F. Toro},
journal = {J. Comput. Phys.},
month = {jan},
number = {4},
pages = {24812513},
title = {{Solvers for the HighOrder Riemann Problem for Hyperbolic Balance Laws}},
volume = {227},
year = {2008},
language = {en},
doi = {10.1016/j.jcp.2007.11.013},
}
 EndNote

%0 Journal Article
%A Castro, CristÃ³bal E.
%A Toro, Eleuterio F.
%D 2008
%N 4
%V 227
%J J. Comput. Phys.
%P 24812513
%T Solvers for the HighOrder Riemann Problem for Hyperbolic Balance Laws
%8 jan