Kinetic Flux Vector Splitting Schemes for Ideal Quantum Gas Dynamics

Novel quantum kinetic flux vector splitting schemes are presented for the computation of ideal quantum gas dynamical flows. The quantum Boltzmann equation approach is adopted and both Bose-Einstein and Fermi-Dirac gases are considered. Formulas for one spatial dimension are first derived, and the resulting kinetic flux vector splitting scheme is tested for shock tube flows. A modified flux vector splitting based on the BGK model is also presented to reduce the numerical diffusion. Implementation of the efficient weighted essentially nonoscillatory method to yield a class of high resolution methods is also devised. Both the classical limit and the nearly degenerate limit are computed. The flow structures can all be accurately captured by the present kinetic flux vector splitting schemes. Results for one-dimensional boson gas under external potential are also included.

[1]  G. Hooyman,et al.  On the Bose-Einstein condensation , 1950, Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences.

[2]  C. Pethick,et al.  Bose–Einstein Condensation in Dilute Gases: Contents , 2008 .

[3]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[4]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[5]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[6]  David E. Miller,et al.  Quantum Statistical Mechanics , 2002 .

[7]  Taku Ohwada,et al.  On the Construction of Kinetic Schemes , 2002 .

[8]  R. H. Sanders,et al.  The possible relation of the 3-kiloparsec arm to explosions in the galactic nucleus , 1974 .

[9]  Jaw-Yen Yang,et al.  A Kinetic Beam Scheme for Relativistic Gas Dynamics , 1997 .

[10]  Anomalous behavior of ideal Fermi gas below two dimensions , 2002, cond-mat/0212465.

[11]  Pierre Degond,et al.  Quantum Moment Hydrodynamics and the Entropy Principle , 2003 .

[12]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[13]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[14]  V. C. Aguilera-Navarro,et al.  Bose-Einstein condensation for general dispersion relations , 1999 .

[15]  S. M. Deshpande,et al.  A second-order accurate kinetic-theory-based method for inviscid compressible flows , 1986 .

[16]  Kun Xu,et al.  Gas-kinetic schemes for unsteady compressible flow simulations , 1998 .

[17]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[18]  Kun Xu,et al.  Numerical Navier-Stokes solutions from gas kinetic theory , 1994 .

[19]  Kun Xu,et al.  A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method , 2001 .

[20]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases , 1939 .

[21]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[22]  J. C. Mandal,et al.  KINETIC FLUX VECTOR SPLITTING FOR EULER EQUATIONS , 1994 .

[23]  M. A. Cayless.  Statistical Mechanics (2nd edn) , 1977 .

[24]  D. Pullin,et al.  Direct simulation methods for compressible inviscid ideal-gas flow , 1980 .

[25]  D. Baganoff,et al.  Kinetic Flux-Vector Splitting for the Navier-Stokes Equations , 1997 .

[26]  Rolf D. Reitz,et al.  One-dimensional compressible gas dynamics calculations using the Boltzmann equation , 1981 .

[27]  E. A. Uehling,et al.  Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I , 1933 .

[28]  Kun Xu,et al.  Dissipative mechanism in Godunov‐type schemes , 2001 .

[29]  Yu-Hsin Shi,et al.  A kinetic beam scheme for ideal quantum gas dynamics , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[30]  Sauro Succi,et al.  Numerical methods for atomic quantum gases with applications to Bose–Einstein condensates and to ultracold fermions , 2004 .

[31]  Alejandro L. Garcia,et al.  Direct simulation Monte Carlo method for the Uehling-Uhlenbeck-Boltzmann equation. , 2003 .

[32]  W. Ketterle,et al.  Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions. , 1996, Physical review. A, Atomic, molecular, and optical physics.