A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws. (English) Zbl 1005.65111

Summary: We consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method. We propose a new formulation of the first-order linear, optimal positive \(N\) scheme, relying on a contour integration of the convective fluxes over the boundaries of an element. Full conservation is achieved for arbitrary flux functions, while the robustness and the monotone shock capturing of the original \(N\) scheme is retained.
The new variant of the \(N\) scheme is combined with the conservative second-order linear \(LDA\) scheme to obtain a nonlinear second-order monotone \(B\) scheme. The performance of the new residual distribution schemes is evaluated on problems governed by the Euler equations. As an application to a more complex system of conservation laws lacking an exact conservative linearization, we solve the ideal magnetohydrodynamics equations in two spatial dimensions.


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76M10 Finite element methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI


[1] Abgrall, R, Toward the ultimate conservative scheme: following the quest, J. comput. phys., 167, 277, (2001) · Zbl 0988.76055
[2] R. Abgrall, and, T. J. Barth, Residual distribution schemes for conservation laws via adaptive quadrature, submitted for publication. · Zbl 1037.35042
[3] D. Caraeni, M. Caraeni, and, L. Fuchs, A Parallel Multidimensional Upwind Algorithm for LES, AIAA Paper 2001-2547 (AIAA Accession number 31057), 2001.
[4] Á. Csı́k, H. Deconinck, and, S. Poedts, Monotone Residual Distribution Schemes for the Ideal 2D Magnetohydrodynamic Equations on Unstructured Grids, AIAA Paper 99-3325 (AIAA Accession number 33525), 1999.
[5] Á. Csı́k, and, H. Deconinck, Space time residual distribution schemes for hyperbolic conservation laws on unstructured linear finite elements, in, Numerical Methods for Fluid Dynamics VII, edited by, M. J. Baines, Oxford University Computing Laboratory, Oxford, 2001. p, 557.
[6] Á. Csı́k, M. Ricchiuto, H. Deconinck, and, S. Poedts, Space-Time Residual Distribution Schemes for Hyperbolic Conservation Laws, AIAA Paper 2001-2617 (AIAA Accession number 31161), 2001.
[7] Csı́k, Á; Deconinck, H; Poedts, S, Monotone residual distribution schemes for the ideal magnetohydrodynamic equations on unstructured grids, Aiaa j., 39, 1532, (2001)
[8] Dai, W; Woodward, P.R, An approximate Riemann solver for ideal magnetohydrodynamics, J. comput. phys., 111, 354, (1994) · Zbl 0797.76052
[9] Deconinck, H; Roe, P.L; Struijs, R, A multidimensional generalization of Roe’s flux difference splitter for the Euler equations, Comput. fluids, 22, 215, (1993) · Zbl 0790.76054
[10] H. Deconinck, K. Sermeus, and, R. Abgrall, Status of Multidimensional Upwind Residual Distribution Schemes and Applications in Aeronautics, AIAA Paper 2000-2328 (AIAA Accession number 33804), 2000. · Zbl 1138.76396
[11] van der Weide, E; Deconinck, H; Issmann, E; Degrez, G, Fluctuation splitting schemes for multidimensional convection problems: an alternative to finite volume and finite element methods, Comput. mech., 23, 199, (1999) · Zbl 0949.76056
[12] S. K. Godunov, Symmetric Form of the Equations of Magnetohydrodynamics, (translated by Timur Linde), available at, http://hpcc.engin.umich.edu/CFD/publications/publications.html.
[13] A. Jameson, Artifical Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence in Transonic and Hypersonic Flows, AIAA Paper 1993-3359, 1993.
[14] H. Nishikawa, M. Rad, and, P. L. Roe, A Third-Order Fluctuation Splitting Scheme That Preserves Potential Flow, AIAA Paper 2001-2595 (AIAA Accession number 31156), 2001.
[15] Paillére, H, multidimensional upwind residual distribution schemes for the Euler and navier – stokes equations on unstructured grids, (1995), Université Libre de Bruxelles · Zbl 0875.76396
[16] K. G. Powell, P. L. Roe, R. S. Myong, T. I. Gombosi, and, D. L. De Zeeuw, An Upwind Scheme for Magnetohydrodynamics, AIAA Paper 1995-1704, 1995.
[17] Powell, K.G; Roe, P.L; Linde, T.J; Gombosi, T.I; De Zeeuw, D.L, A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. comput. phys., 154, 284, (1999) · Zbl 0952.76045
[18] M. Rad, and, P. L. Roe, An Euler code that can compute potential flow, in, Finite Volumes for Complex Applications II-Problems and Perspectives, edited by, R. Vilsmeier, F. Benkhaldoun, D. Hanel, proceedings published by, Hermes, Duisberg, Germany, 1999. · Zbl 1052.65556
[19] P. L. Roe, and, H. Nishikawa, Adaptive grid generation by minimising residuals, in, Numerical Methods for Fluid Dynamics VII, edited by, M. J. Baines, Oxford University Computing Laboratory, Oxford, p, 45, 2001.
[20] Struijs, R, A multi-dimensional upwind discretization method for the Euler equations on unstructured grids, (1994), University of Delft
[21] Toth, G, The ∇·B=0 constraint in shock-capturing magnetohydrodynamics codes, J. comput. phys., 161, 605, (2000) · Zbl 0980.76051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.