# On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Tim Kröger; Sebastian Noelle; Susanne Zimmermann^{[1]}

- [1] ETH Zentrum, Seminar für Angewandte Mathematik, 8092 Zürich, Switzerland.

- Volume: 38, Issue: 6, page 989-1009
- ISSN: 0764-583X

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topKröger, Tim, Noelle, Sebastian, and 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 38.6 (2004): 989-1009. <http://eudml.org/doc/246051>.

@article{Kröger2004,

abstract = {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.},

affiliation = {ETH Zentrum, Seminar für Angewandte Mathematik, 8092 Zürich, Switzerland.},

author = {Kröger, Tim, Noelle, Sebastian, Zimmermann, Susanne},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {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; Galerkin method; method of transport},

language = {eng},

number = {6},

pages = {989-1009},

publisher = {EDP-Sciences},

title = {On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws},

url = {http://eudml.org/doc/246051},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Kröger, Tim

AU - Noelle, Sebastian

AU - Zimmermann, Susanne

TI - On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 6

SP - 989

EP - 1009

AB - 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.

LA - eng

KW - 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; Galerkin method; method of transport

UR - http://eudml.org/doc/246051

ER -

## References

top- [1] 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. Zbl0957.82028
- [2] Y. Brenier, Average multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21 (1984) 1013–1037. Zbl0565.65054
- [3] 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. Zbl0099.41501
- [4] C. Cercignani, The Boltzmann equation and its applications. Springer-Verlag, New York (1988). Zbl0646.76001MR1313028
- [5] R. Courant and D. Hilbert, Methods of Mathematical Physics II. Interscience Publishers, New York (1962). Zbl0099.29504
- [6] S.M. Deshpande, A second-order accurate kinetic-theory-based method for inviscid compressible flows. NASA Technical Paper 2613 (1986).
- [7] 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. Zbl0790.76054
- [8] M. Fey, Ein echt mehrdimensionales Verfahren zur Lösung der Eulergleichungen. Dissertation, ETH Zürich, Switzerland (1993).
- [9] M. Fey, Multidimensional upwinding. I. The method of transport for solving the Euler equations. J. Comput. Phys. 143 (1998) 159–180. Zbl0932.76050
- [10] M. Fey, Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998) 181–199. Zbl0932.76051
- [11] 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. Zbl1052.65533
- [12] E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York (1996). Zbl0860.65075MR1410987
- [13] A. Jeffrey and T. Taniuti, Non-linear wave propagation. Academic Press, New York (1964). Zbl0117.21103MR167137
- [14] M. Junk, A kinetic approach to hyperbolic systems and the role of higher order entropies. Int. Ser. Numer. Math. 140, 141 (2001) 583–592.
- [15] T. Kröger, Multidimensional systems of hyperbolic conservation laws, numerical schemes, and characteristic theory. Dissertation, RWTH Aachen, Germany (2004).
- [16] 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). Zbl1077.35007
- [17] D. Kröner, Numerical schemes for conservation laws. Wiley Teubner, Stuttgart (1997). Zbl0872.76001MR1437144
- [18] R.J. LeVeque, Numerical methods for conservation laws. Birkhäuser, Basel (1990). Zbl0723.65067MR1077828
- [19] 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. Zbl0880.65079
- [20] 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. Zbl0951.35076
- [21] 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. Zbl1078.65562
- [22] M. Lukáčová-Medviďová, J. Saibertová and G. Warnecke, Finite volume evolution Galerkin methods for nonlinear hyperbolic systems. J. Comp. Phys. 183 (2002) 533–562. Zbl1090.65536
- [23] 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. Zbl0967.65100
- [24] S. Ostkamp, Multidimensional Characteristic Galerkin Schemes and Evolution Operators for Hyperbolic Systems. Dissertation, Hannover University, Germany (1995). Zbl0831.76067MR1361170
- [25] S. Ostkamp, Multidimensional characteristic Galerkin methods for hyperbolic systems. Math. Meth. Appl. Sci. 20 (1997) 1111–1125. Zbl0880.35065
- [26] B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405–1421. Zbl0714.76078
- [27] B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29 (1992) 1–19. Zbl0744.76088
- [28] P. Prasad, Nonlinear hyperbolic waves in multi-dimensions. Chapman & Hall/CRC, New York (2001). Zbl0992.35001MR1852712
- [29] J. Quirk, A contribution to the great Riemann solver debate. Int. J. Numer. Meth. Fluids 18 (1994) 555–574. Zbl0794.76061
- [30] P. Roe, Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics. J. Comput. Phys. 63 (1986) 458–476. Zbl0587.76126
- [31] 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. Zbl0468.76066
- [32] C.v. Törne, MOTICE – Adaptive, Parallel Numerical Solution of Hyperbolic Conservation Laws. Dissertation, Bonn University, Germany. Bonner Mathematische Schriften, No. 334 (2000). Zbl0971.76001
- [33] E. Toro, Riemann solvers and numerical methods for fluid dynamics. Second edition, Springer, Berlin (1999). Zbl0801.76062MR1717819
- [34] K. Xu, Gas-kinetic schemes for unsteady compressible flow simulations. Lect. Ser. Comp. Fluid Dynamics, VKI report 1998-03 (1998).
- [35] S. Zimmermann, The method of transport for the Euler equations written as a kinetic scheme. Int. Ser. Numer. Math. 141 (2001) 999–1008. Zbl0929.35118
- [36] S. Zimmermann, Properties of the Method of Transport for the Euler Equations. Dissertation, ETH Zürich, Switzerland (2001).

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