June  2011, 4(2): 517-530. doi: 10.3934/krm.2011.4.517

On kinetic flux vector splitting schemes for quantum Euler equations

1. 

Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, United States

2. 

Department of Mathematics, University of Wisconsin, Madison, WI 53706

Received  November 2010 Published  April 2011

The kinetic flux vector splitting (KFVS) scheme, when used for quantum Euler equations, as was done by Yang et al [22], requires the integration of the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution), giving a numerical flux much more complicated than the classical counterpart. As a result, a nonlinear 2 by 2 system that connects the macroscopic quantities temperature and fugacity with density and internal energy needs to be inverted by iterative methods at every spatial point and every time step. In this paper, we propose to use a simple classical KFVS scheme for the quantum hydrodynamics based on the key observation that the quantum and classical Euler equations share the same form if the (quantum) internal energy rather than temperature is used in the flux. This motivates us to use a classical Maxwellian - that depends on the internal energy rather than temperature - instead of the quantum one in the construction of the scheme, yielding a KFVS which is purely classical. This greatly simplifies the numerical algorithm and reduces the computational cost. The proposed schemes are tested on a 1-D shock tube problem for the Bose and Fermi gases in both classical and nearly degenerate regimes.
Citation: Jingwei Hu, Shi Jin. On kinetic flux vector splitting schemes for quantum Euler equations. Kinetic & Related Models, 2011, 4 (2) : 517-530. doi: 10.3934/krm.2011.4.517
References:
[1]

V. V. Aristov and F. G. Tcheremissine, Kinetic numerical method for rarefied and continuum gas flows,, in, 1 (1985), 269.

[2]

C. Cercignani, "The Boltzmann Equation and Its Applications,", Springer-Verlag, (1988).

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", 3rd edition, (1990).

[4]

S. M. Deshpande, Kinetic theory based new upwind methods for inviscid compressible flows,, AIAA Paper 86-0275, (1986), 86.

[5]

S. M. Deshpande and R. Raul, "Kinetic Theory Based Fluid-in-Cell Method for Eulerian Fluid Dynamics,", Report 82 FM 14, (1982).

[6]

T. G. Elizarova and B. N. Chetverushkin, Kinetic-consistent finite-difference gas dynamic schemes,, Japan Soc. Comput. Fluid Dyn., (1989), 501.

[7]

A. Harten, P. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,, SIAM Rev., 25 (1983), 35. doi: 10.1137/1025002.

[8]

S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions,, Commun. Pure Appl. Math., 48 (1995), 235. doi: 10.1002/cpa.3160480303.

[9]

R. J. LeVeque, "Numerical Methods for Conservation Laws,", 2nd edition, (1992).

[10]

L. W. Nordheim, On the kinetic method in the new statistics and its application in the electron theory of conductivity,, Proc. R. Soc. London, 119 (1928), 689. doi: 10.1098/rspa.1928.0126.

[11]

R. K. Pathria, "Statistical Mechanics,", 2$^{nd}$ edition, (1996).

[12]

B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property,, SIAM J. Numer. Anal., 27 (1990), 1405. doi: 10.1137/0727081.

[13]

B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions,, SIAM J. Numer. Anal., 29 (1992), 1. doi: 10.1137/0729001.

[14]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes: The Art of Scientific Computing,", 3$^{rd}$ edition, (2007).

[15]

D. I. Pullin, Direct simulation methods for compressible inviscid ideal-gas flow,, J. Comput. Phys., 34 (1980), 231. doi: 10.1016/0021-9991(80)90107-2.

[16]

R. D. Reitz, One-dimensional compressible gas dynamics calculations using the Boltzmann equation,, J. Comput. Phys., 42 (1981), 108. doi: 10.1016/0021-9991(81)90235-7.

[17]

J. Ross and J. G. Kirkwood, The statistical-mechanical theory of transport processes. VIII. Quantum theory of transport in gases,, J. Chem. Phys., 22 (1954), 1094. doi: 10.1063/1.1740271.

[18]

A. Sommerfeld, Zur elektronentheorie der metalle auf grund der Fermischen statistik,, Zeitschrift für Physik A Hadrons and Nuclei, 47 (1928), 1.

[19]

E. A. Uehling, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. II,, Phys. Rev., 46 (1934), 917. doi: 10.1103/PhysRev.46.917.

[20]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I,, Phys. Rev., 43 (1933), 552. doi: 10.1103/PhysRev.43.552.

[21]

B. Van Leer, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme,, J. Comput. Phys., 14 (1974), 361. doi: 10.1016/0021-9991(74)90019-9.

[22]

J. Y. Yang, T. Y. Hsieh and Y. H. Shi, Kinetic flux vector splitting schemes for ideal quantum gas dynamics,, SIAM J. Sci. Comput., 29 (2007), 221. doi: 10.1137/050646664.

[23]

J. Y. Yang, T. Y. Hsieh, Y. H. Shi and K. Xu, High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics,, J. Comput. Phys., 227 (2007), 967. doi: 10.1016/j.jcp.2007.08.014.

show all references

References:
[1]

V. V. Aristov and F. G. Tcheremissine, Kinetic numerical method for rarefied and continuum gas flows,, in, 1 (1985), 269.

[2]

C. Cercignani, "The Boltzmann Equation and Its Applications,", Springer-Verlag, (1988).

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", 3rd edition, (1990).

[4]

S. M. Deshpande, Kinetic theory based new upwind methods for inviscid compressible flows,, AIAA Paper 86-0275, (1986), 86.

[5]

S. M. Deshpande and R. Raul, "Kinetic Theory Based Fluid-in-Cell Method for Eulerian Fluid Dynamics,", Report 82 FM 14, (1982).

[6]

T. G. Elizarova and B. N. Chetverushkin, Kinetic-consistent finite-difference gas dynamic schemes,, Japan Soc. Comput. Fluid Dyn., (1989), 501.

[7]

A. Harten, P. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,, SIAM Rev., 25 (1983), 35. doi: 10.1137/1025002.

[8]

S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions,, Commun. Pure Appl. Math., 48 (1995), 235. doi: 10.1002/cpa.3160480303.

[9]

R. J. LeVeque, "Numerical Methods for Conservation Laws,", 2nd edition, (1992).

[10]

L. W. Nordheim, On the kinetic method in the new statistics and its application in the electron theory of conductivity,, Proc. R. Soc. London, 119 (1928), 689. doi: 10.1098/rspa.1928.0126.

[11]

R. K. Pathria, "Statistical Mechanics,", 2$^{nd}$ edition, (1996).

[12]

B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property,, SIAM J. Numer. Anal., 27 (1990), 1405. doi: 10.1137/0727081.

[13]

B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions,, SIAM J. Numer. Anal., 29 (1992), 1. doi: 10.1137/0729001.

[14]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes: The Art of Scientific Computing,", 3$^{rd}$ edition, (2007).

[15]

D. I. Pullin, Direct simulation methods for compressible inviscid ideal-gas flow,, J. Comput. Phys., 34 (1980), 231. doi: 10.1016/0021-9991(80)90107-2.

[16]

R. D. Reitz, One-dimensional compressible gas dynamics calculations using the Boltzmann equation,, J. Comput. Phys., 42 (1981), 108. doi: 10.1016/0021-9991(81)90235-7.

[17]

J. Ross and J. G. Kirkwood, The statistical-mechanical theory of transport processes. VIII. Quantum theory of transport in gases,, J. Chem. Phys., 22 (1954), 1094. doi: 10.1063/1.1740271.

[18]

A. Sommerfeld, Zur elektronentheorie der metalle auf grund der Fermischen statistik,, Zeitschrift für Physik A Hadrons and Nuclei, 47 (1928), 1.

[19]

E. A. Uehling, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. II,, Phys. Rev., 46 (1934), 917. doi: 10.1103/PhysRev.46.917.

[20]

E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I,, Phys. Rev., 43 (1933), 552. doi: 10.1103/PhysRev.43.552.

[21]

B. Van Leer, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme,, J. Comput. Phys., 14 (1974), 361. doi: 10.1016/0021-9991(74)90019-9.

[22]

J. Y. Yang, T. Y. Hsieh and Y. H. Shi, Kinetic flux vector splitting schemes for ideal quantum gas dynamics,, SIAM J. Sci. Comput., 29 (2007), 221. doi: 10.1137/050646664.

[23]

J. Y. Yang, T. Y. Hsieh, Y. H. Shi and K. Xu, High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics,, J. Comput. Phys., 227 (2007), 967. doi: 10.1016/j.jcp.2007.08.014.

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