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Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems

Summary: Essentially non-oscillatory Residual Distribution schemes
for hyperbolic problems
R. Abgrall *
Institut Universitaire de France and Mathe´matiques Applique´es de Bordeaux, Projet Scalapplix, INRIA FutURs,
Universite´ Bordeaux I, 351 cours de la Libe´ration, F-33405 Talence Cedex, France
Received 3 June 2005; received in revised form 22 September 2005; accepted 14 October 2005
Available online 3 January 2006
The Residual Distribution (RD) schemes are an alternative to standard high order accurate finite volume schemes. They
have several advantages: a better accuracy, a much more compact stencil, easy parallelization. However, they face several
problems, at least for steady problems which are the only cases considered here. The solution is obtained via an iterative
method. The iterative convergence must be good in order to get spatially accurate solutions, as suggested by the few the-
oretical results available for the RD schemes. In many cases, especially for systems, the iterative convergence is not suf-
ficient to guaranty the theoretical accuracy. In fact, up to our knowledge, the iterative convergence is correct in only
two cases: for first order monotone schemes and the (scalar) Struij's PSI scheme which is a multidimensional upwind
scheme. Up to our knowledge, the iterative convergence is poor for systems, except for the blended scheme of Deconinck
et al. [A´ . Csi´k, M. Ricchiuto, H. Deconinck, A conservative formulation of the multidimensional upwind residual distri-
bution schemes for general nonlinear conservation laws, J. Comput. Phys. 179(2) (2002) 286­312] and Abgrall [R. Abgrall,
Toward the ultimate conservative scheme: following the quest, J. Comput. Phys. 167(2) (2001) 277­315] which are also a
genuinely multidimensional upwind scheme.


Source: Abgrall, Rémi - Institut de Mathematiques de Bordeaux, Université Bordeaux


Collections: Mathematics