On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 6, pp. 989-1009.

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp's equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.

DOI : https://doi.org/10.1051/m2an:2004047
Classification : 35C15,  35L65,  65D32,  65M25,  76M12,  76N15,  76P05
Mots clés : systems of conservation laws, Fey's method of transport, Euler equations, Boltzmann equation, kinetic schemes, bicharacteristic theory, state decompositions, flux decompositions, exact and approximate integral representations, quadrature rules
@article{M2AN_2004__38_6_989_0,
author = {Kr\"oger, Tim and Noelle, Sebastian and Zimmermann, Susanne},
title = {On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {989--1009},
publisher = {EDP-Sciences},
volume = {38},
number = {6},
year = {2004},
doi = {10.1051/m2an:2004047},
zbl = {1083.35063},
mrnumber = {2108941},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004047/}
}
Kröger, Tim; Noelle, Sebastian; Zimmermann, Susanne. On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 6, pp. 989-1009. doi : 10.1051/m2an:2004047. http://www.numdam.org/articles/10.1051/m2an:2004047/

 F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113-170. | Zbl 0957.82028

 Y. Brenier, Average multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21 (1984) 1013-1037. | Zbl 0565.65054

 D.S. Butler, The numerical solution of hyperbolic systems of partial differential equations in three independent variables, in Proc. Roy. Soc. 255A (1960) 232-252. | Zbl 0099.41501

 C. Cercignani, The Boltzmann equation and its applications. Springer-Verlag, New York (1988). | MR 1313028 | Zbl 0646.76001

 R. Courant and D. Hilbert, Methods of Mathematical Physics II. Interscience Publishers, New York (1962). | Zbl 0099.29504

 S.M. Deshpande, A second-order accurate kinetic-theory-based method for inviscid compressible flows. NASA Technical Paper 2613 (1986).

 H. Deconinck, P.L. Roe and R. Struijs, A multidimensional generalization of Roe's flux difference splitter for the Euler equations. Comput. Fluids 22 (1993) 215-222. | Zbl 0790.76054

 M. Fey, Ein echt mehrdimensionales Verfahren zur Lösung der Eulergleichungen. Dissertation, ETH Zürich, Switzerland (1993).

 M. Fey, Multidimensional upwinding. I. The method of transport for solving the Euler equations. J. Comput. Phys. 143 (1998) 159-180. | Zbl 0932.76050

 M. Fey, Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998) 181-199. | Zbl 0932.76051

 M. Fey, S. Noelle and C.v. Törne, The MoT-ICE: a new multi-dimensional wave-propagation-algorithm based on Fey's method of transport. With application to the Euler- and MHD-equations. Int. Ser. Numer. Math. 140, 141 (2001) 373-380. | Zbl 1052.65533

 E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York (1996). | MR 1410987 | Zbl 0860.65075

 A. Jeffrey and T. Taniuti, Non-linear wave propagation. Academic Press, New York (1964). | MR 167137

 M. Junk, A kinetic approach to hyperbolic systems and the role of higher order entropies. Int. Ser. Numer. Math. 140, 141 (2001) 583-592.

 T. Kröger, Multidimensional systems of hyperbolic conservation laws, numerical schemes, and characteristic theory. Dissertation, RWTH Aachen, Germany (2004).

 T. Kröger and S. Noelle, Numerical comparison of the method of transport to a standard scheme. Comp. Fluids (2004) (doi: 10.1016/j.compfluid.2003.12.002) (in print). | Zbl 1077.35007

 D. Kröner, Numerical schemes for conservation laws. Wiley Teubner, Stuttgart (1997). | MR 1437144 | Zbl 0872.76001

 R.J. Leveque, Numerical methods for conservation laws. Birkhäuser, Basel (1990). | MR 1077828 | Zbl 0723.65067

 P. Lin, K.W. Morton and E. Süli, Characteristic Galerkin schemes for scalar conservation laws in two and three space dimensions. SIAM J. Numer. Anal. 34 (1997) 779-796. | Zbl 0880.65079

 M. Lukáčová-Medviďová, K.W. Morton and G. Warnecke, Evolution Galerkin methods for hyperbolic systems in two space dimensions. Math. Comp. 69 (2000) 1355-1384. | Zbl 0951.35076

 M. Lukáčová-Medviďová, K.W. Morton and G. Warnecke, Finite volume evolution Galerkin (FVEG) methods hyperbolic systems. SIAM J. Sci. Comp. 26 (2004) 1-30. | Zbl 1078.65562

 M. Lukáčová-Medviďová, J. Saibertová and G. Warnecke, Finite volume evolution Galerkin methods for nonlinear hyperbolic systems. J. Comp. Phys. 183 (2002) 533-562. | Zbl 1090.65536

 S. Noelle, The MoT-ICE: a new high-resolution wave-propagation algorithm for multidimensional systems of conservation laws based on Fey's Method of Transport. J. Comput. Phys. 164 (2000) 283-334. | Zbl 0967.65100

 S. Ostkamp, Multidimensional Characteristic Galerkin Schemes and Evolution Operators for Hyperbolic Systems. Dissertation, Hannover University, Germany (1995). | MR 1361170 | Zbl 0831.76067

 S. Ostkamp, Multidimensional characteristic Galerkin methods for hyperbolic systems. Math. Meth. Appl. Sci. 20 (1997) 1111-1125. | Zbl 0880.35065

 B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405-1421. | Zbl 0714.76078

 B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29 (1992) 1-19. | Zbl 0744.76088

 P. Prasad, Nonlinear hyperbolic waves in multi-dimensions. Chapman & Hall/CRC, New York (2001). | MR 1852712 | Zbl 0992.35001

 J. Quirk, A contribution to the great Riemann solver debate. Int. J. Numer. Meth. Fluids 18 (1994) 555-574. | Zbl 0794.76061

 P. Roe, Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics. J. Comput. Phys. 63 (1986) 458-476. | Zbl 0587.76126

 J.L. Steger and R.F. Warming, Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J. Comput. Phys. 40 (1981) 263-293. | Zbl 0468.76066

 C.v. Törne, MOTICE - Adaptive, Parallel Numerical Solution of Hyperbolic Conservation Laws. Dissertation, Bonn University, Germany. Bonner Mathematische Schriften, No. 334 (2000). | Zbl 0971.76001

 E. Toro, Riemann solvers and numerical methods for fluid dynamics. Second edition, Springer, Berlin (1999). | MR 1717819 | Zbl 0801.76062

 K. Xu, Gas-kinetic schemes for unsteady compressible flow simulations. Lect. Ser. Comp. Fluid Dynamics, VKI report 1998-03 (1998).

 S. Zimmermann, The method of transport for the Euler equations written as a kinetic scheme. Int. Ser. Numer. Math. 141 (2001) 999-1008. | Zbl 0929.35118

 S. Zimmermann, Properties of the Method of Transport for the Euler Equations. Dissertation, ETH Zürich, Switzerland (2001).