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Journal of Scientific Computing

Erschienen in:

01.04.2013

A Characteristics Based Genuinely Multidimensional Discrete Kinetic Scheme for the Euler Equations

verfasst von: K. R. Arun, M. Lukáčová-Medviďová

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2013

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Abstract

In this paper we present the results of a kinetic relaxation scheme for an arbitrary hyperbolic system of conservation laws in two space dimensions. We propose a new discrete velocity Boltzmann equation, which is an improvement over the previous models in terms of the isotropic coverage of the multidimensional domain by the foot of the characteristic. The discrete kinetic equation is solved by a splitting method consisting of a convection step and a collision step. The convection step involves only the solution of linear transport equations whereas the collision step instantaneously relaxes the distribution function to a local Maxwellian. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. Finally some numerical results are presented which confirm the robustness and correct multidimensional behaviour of the proposed scheme.

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Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000) MathSciNetCrossRefMATH

Arun, K.R., Kraft, M., Lukáčová-Medviďová, M., Prasad, P.: Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions. J. Comput. Phys. 228, 565–590 (2009) MathSciNetCrossRefMATH

Arun, K.R., Lukáčová-Medviďová, M., Prasad, P., Raghurama Rao, S.V.: A second order accurate kinetic relaxation scheme for inviscid compressible flows. Technical report, Department of Mathematics, Indian Institute of Science, Bangalore, India (2010). Available from http://math.iisc.ernet.in/~prasad/

Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954) CrossRefMATH

Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95, 113–170 (1999) MathSciNetCrossRefMATH

Cercignani, C.: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer, New York (1988) CrossRefMATH

Chen, G.Q., Levermore, C.D., Liu, T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994) MathSciNetCrossRefMATH

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

Deshpande, S.M.: Kinetic flux splitting schemes. In: Hafez, M., Oshima, K. (eds.) Computational Fluid Dynamics Review 1995: A State-of-the-Art Reference to the Latest Developments in CFD. Wiley, New York (1995)

10.

Fey, M.: Multidimensional upwinding. I. The method of transport for solving the Euler equations. J. Comput. Phys. 143, 159–180 (1998) MathSciNetCrossRefMATH

11.

Fey, M.: Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143, 181–199 (1998) MathSciNetCrossRefMATH

12.

Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118. Springer, New York (1996) MATH

13.

Jameson, A., Schmidt, W., Turkel, E.: Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes. AIAA Paper 81-1259 (1981)

14.

Jin, S.: Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122, 51–67 (1995) MathSciNetCrossRefMATH

15.

Jin, S., Xin, Z.P.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276 (1995) MathSciNetCrossRefMATH

16.

Kunik, M., Qamar, S., Warnecke, G.: Second-order accurate kinetic schemes for the ultra-relativistic Euler equations. J. Comput. Phys. 192, 695–726 (2003) MathSciNetCrossRefMATH

17.

Langseth, J.O., LeVeque, R.J.: A wave propagation method for three-dimensional hyperbolic conservation laws. J. Comput. Phys. 165, 126–166 (2000) MathSciNetCrossRefMATH

18.

LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002) CrossRefMATH

19.

LeVeque, R.J., Walder, R.: Grid alignment effects and rotated methods for computing complex flows in astrophysics. In: Proceedings of the Ninth GAMM-Conference on Numerical Methods in Fluid Mechanics, Lausanne, 1991. Notes Numer. Fluid Mech., vol. 35, pp. 376–385. Vieweg, Wiesbaden (1992)

20.

Liu, T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175 (1987) CrossRefMATH

21.

Lukáčová-Medviďová, M., Morton, K.W., Warnecke, G.: Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems. SIAM J. Sci. Comput. 26, 1–30 (2004) MathSciNetCrossRefMATH

22.

Natalini, R.: Recent results on hyperbolic relaxation problems. In: Analysis of Systems of Conservation Laws, Aachen, 1997. Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99, pp. 128–198. Chapman & Hall/CRC, Boca Raton (1999)

23.

Noelle, S.: 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, 283–334 (2000) MathSciNetCrossRefMATH

24.

Noelle, S.W.C.: On the limits of operator splitting: numerical experiments for the complex Burgers equation. Technical report tr-91-03, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (1991)

25.

Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005) MathSciNetMATH

26.

Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27, 1405–1421 (1990) MathSciNetCrossRefMATH

27.

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

28.

Roe, P.L.: Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics. J. Comput. Phys. 63, 458–476 (1986) MathSciNetCrossRefMATH

29.

Shubin, G.R., Bell, J.B.: An analysis of the grid orientation effect in numerical simulation of miscible displacement. Comput. Methods Appl. Mech. Eng. 47, 47–71 (1984) CrossRefMATH

30.

Tadmor, E.: Approximate solutions of nonlinear conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Cetraro, 1997. Lecture Notes in Math., vol. 1697, pp. 1–149. Springer, Berlin (1998) CrossRef

31.

Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984) MathSciNetCrossRefMATH

32.

Xu, K.: Gas kinetic scheme for unsteady compressible flow simulations. In: Lecture Note Series 1998-03. Von Kárman Institute for Fluid Dynamics, Rhode Saint Genèse (1998)

Titel: A Characteristics Based Genuinely Multidimensional Discrete Kinetic Scheme for the Euler Equations
verfasst von: K. R. Arun
M. Lukáčová-Medviďová
Publikationsdatum: 01.04.2013
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2013
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-012-9623-6

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