Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems

Abstract

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 theoretical results available for the RD schemes. In many cases, especially for systems, the iterative convergence is not sufficient 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. [Á. Csík, M. Ricchiuto, H. Deconinck, A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws, J. Comput. Phys. 179(2) (2002) 286–312] and Abgrall [R. Abgrall, Toward the ultimate conservative scheme: following the quest, J. Comput. Phys. 167(2) (2001) 277–315] which are also a genuinely multidimensional upwind scheme. A second drawback is that their construction relies, up to now, on a single first order scheme: the N scheme. However, it is known that standard first order finite volume schemes can be rephrased into a Residual Distribution framework. Unfortunately, the standard way of upgrading the order of accuracy to second order leads to very unsatisfactory results but clearly the construction of good schemes based on a wider class of first order schemes would be interesting. In this paper, we analyze these two problems, and show they are linked. We propose a fix and demonstrate its efficiency on several test cases that cover a wide range of applications. Our solution extends considerably the number of working RD schemes.