A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Filbet, Francis; Hu, Jingwei; Jin, Shi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 443-463.

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010) 7625-7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution) at each time step and every mesh point, one has to invert a nonlinear equation that connects the macroscopic quantity fugacity with density and internal energy. Setting a good initial guess for the iterative method is troublesome in most cases because of the complexity of the quantum functions (Bose-Einstein or Fermi-Dirac function). In this paper, we propose to penalize the quantum collision term by a ‘classical' BGK operator instead of the quantum one. This is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. The scheme so designed avoids the aforementioned difficulty, and one can show that the density distribution is still driven toward the quantum equilibrium. Numerical results are presented to illustrate the efficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also develop a spectral method for the quantum collision operator.

[1] L. Arlotti and M. Lachowicz, Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations. J. Math. Phys. 38 (1997) 3571-3588. | MR | Zbl

[2] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann. Acta Math. 60 (1933) 91-146. | JFM | MR

[3] C. Cercignani, The Boltzmann Equation and Its Applications. Springer-Verlag, New York (1988). | MR | Zbl

[4] G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations. arXiv:1010.1472. | MR | Zbl

[5] M. Escobedo and S. Mischler, On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. 80 (2001) 471-515. | MR | Zbl

[6] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229 (2010) 7625-7648. | MR | Zbl

[7] F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in NlogN. SIAM J. Sci. Comput. 28 (2006) 1029-1053. | MR | Zbl

[8] T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-Preserving schemes for kinetic-fluid modeling of disperse two-phase flows. Preprint. | MR

[9] J.W. Hu and S. Jin, On kinetic flux vector splitting schemes for quantum Euler equations. KRM 4 (2011) 517-530. | MR | Zbl

[10] J.W. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator. Preprint. | MR | Zbl

[11] R.J. Leveque, Numerical Methods for Conservation Laws, 2nd edition. Birkhäuser Verlag, Basel (1992). | MR | Zbl

[12] X. Lu, A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long-time behavior. J. Statist. Phys. 98 (2000) 1335-1394. | MR | Zbl

[13] X. Lu, On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles. J. Statist. Phys. 105 (2001) 353-388. | MR | Zbl

[14] X. Lu and B. Wennberg, On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal. 168 (2003) 1-34. | MR | Zbl

[15] P. Markowich and L. Pareschi, Fast, conservative and entropic numerical methods for the Bosonic Boltzmann equation. Numer. Math. 99 (2005) 509-532. | MR | Zbl

[16] C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75 (2006) 1833-1852. | MR | Zbl

[17] L.W. Nordheim, On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. R. Soc. Lond. Ser. A 119 (1928) 689-698. | JFM

[18] L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (2000) 1217-1245. | MR | Zbl

[19] L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129-155. | MR | Zbl

[20] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3th edition. Cambridge University Press, Cambridge (2007). | Zbl

[21] E.A. Uehling and G.E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev. 43 (1933) 552-561. | Zbl

[22] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Mechanics I. edited by S. Friedlander and D. Serre, North-Holland (2002) 71-305. | MR | Zbl