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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

top
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.

How to cite

top

Kröger, 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 38.6 (2010): 989-1009. <http://eudml.org/doc/194252>.

@article{Kröger2010,
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. },
author = {Kröger, Tim, Noelle, Sebastian, Zimmermann, Susanne},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
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; Euler equations},
language = {eng},
month = {3},
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/194252},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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; Euler equations
UR - http://eudml.org/doc/194252
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).  Zbl0646.76001
  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 Paper2613 (1986).  
  7. H. Deconinck, P.L. Roe and R. Struijs, A multidimensional generalization of Roe's flux difference splitter for the Euler equations. Comput. Fluids22 (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.  
  12. E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York (1996).  Zbl0860.65075
  13. A. Jeffrey and T. Taniuti, Non-linear wave propagation. Academic Press, New York (1964).  Zbl0117.21103
  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. M. Junk and J. Struckmeier, Consistency analysis of mesh-free methods for conservation laws, Mitt. Ges. Angew. Math. Mech. 24, No. 2, 99 (2001).  Zbl0994.65093
  16. T. Kröger, Multidimensional systems of hyperbolic conservation laws, numerical schemes, and characteristic theory. Dissertation, RWTH Aachen, Germany (2004).  
  17. T. Kröger and S. Noelle, Numerical comparison of the method of transport to a standard scheme. Comp. Fluids (2004) (doi: ) (in print).  Zbl1077.35007DOI10.1016/j.compfluid.2003.12.002
  18. D. Kröner, Numerical schemes for conservation laws. Wiley Teubner, Stuttgart (1997).  Zbl0872.76001
  19. R.J. LeVeque, Numerical methods for conservation laws. Birkhäuser, Basel (1990).  Zbl0723.65067
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. S. Ostkamp, Multidimensional Characteristic Galerkin Schemes and Evolution Operators for Hyperbolic Systems. Dissertation, Hannover University, Germany (1995).  Zbl0831.76067
  26. S. Ostkamp, Multidimensional characteristic Galerkin methods for hyperbolic systems. Math. Meth. Appl. Sci. 20 (1997) 1111–1125.  Zbl0880.35065
  27. B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27 (1990) 1405–1421.  Zbl0714.76078
  28. 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
  29. P. Prasad, Nonlinear hyperbolic waves in multi-dimensions. Chapman & Hall/CRC, New York (2001).  
  30. J. Quirk, A contribution to the great Riemann solver debate. Int. J. Numer. Meth. Fluids18 (1994) 555–574.  Zbl0794.76061
  31. P. Roe, Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics. J. Comput. Phys. 63 (1986) 458–476.  Zbl0587.76126
  32. 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
  33. C.v. Törne, MOTICE – Adaptive, Parallel Numerical Solution of Hyperbolic Conservation Laws. Dissertation, Bonn University, Germany. Bonner Mathematische Schriften, No. 334 (2000).  
  34. E. Toro, Riemann solvers and numerical methods for fluid dynamics. Second edition, Springer, Berlin (1999).  Zbl0923.76004
  35. K. Xu, Gas-kinetic schemes for unsteady compressible flow simulations. Lect. Ser. Comp. Fluid Dynamics, VKI report 1998-03 (1998).  
  36. 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
  37. S. Zimmermann, Properties of the Method of Transport for the Euler Equations. Dissertation, ETH Zürich, Switzerland (2001).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.