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Numerical schemes for kinetic equations in the anomalous diffusion limit. I: The case of heavy-tailed equilibrium. (English) Zbl 1382.65238

Summary: In this work, we propose some numerical schemes for linear kinetic equations in the anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation. Our aim is to develop numerical schemes for the original kinetic model which works for the different regimes, without being restricted by stability conditions of standard explicit time integrators. Starting from some numerical schemes for the diffusion asymptotics, their extension to the anomalous diffusion limit is then studied. In this case, it is crucial to capture the effect of the large velocities of the heavy-tailed equilibrium, so that some important transformations of the schemes derived for the diffusion asymptotics are needed. As a result, we obtain numerical schemes which enjoy the asymptotic preserving property in the anomalous diffusion limit; that is, they do not suffer from the restriction on the time step, and they degenerate towards the fractional diffusion limit when the mean free path goes to zero. We also numerically investigate the uniform accuracy and construct a class of numerical schemes satisfying this property. Finally, the efficiency of the different numerical schemes is shown through numerical experiments.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35F05 Linear first-order PDEs
82D75 Nuclear reactor theory; neutron transport
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[1] N. Ben Abdallah, A. Mellet, and M. Puel, {\it Anomalous diffusion limit for kinetic equations with degenerate collision frequency}, Math. Models Methods Appl. Sci., 21 (2011), pp. 2249-2262, http://dx.doi.org/10.1142/S0218202511005738 doi:10.1142/S0218202511005738. · Zbl 1331.76106
[2] N. Ben Abdallah, A. Mellet, and M. Puel, {\it Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach}, Kinet. Relat. Models, 4 (2011), pp. 873-900, http://dx.doi.org/10.3934/krm.2011.4.873 doi:10.3934/krm.2011.4.873. · Zbl 1242.76304
[3] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, {\it Boundary layers and homogenization of transport processes}, Publ. Res. Int. Math. Sci., 15 (1979), pp. 53-157, http://dx.doi.org/10.2977/prims/1195188427 doi:10.2977/prims/1195188427. · Zbl 0408.60100
[4] A.V. Bobylev, J.A. Carrillo, and I.M. Gamba, {\it On some properties of kinetic and hydrodynamic equations for inelastic interactions}, J. Statist. Phys., 98 (2000), pp. 743-773, http://dx.doi.org/10.1023/A:1018627625800 doi:10.1023/A:1018627625800. · Zbl 1056.76071
[5] A.V. Bobylev and I.M. Gamba, {\it Boltzmann equations for mixtures of Maxwell gases: Exact solutions and power like tails}, J. Stat. Phys., 124 (2006), pp. 497-516, http://dx.doi.org/10.1007/s10955-006-9044-8 doi:10.1007/s10955-006-9044-8. · Zbl 1134.82032
[6] C. Buet and S. Cordier, {\it Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models}, C. R. Math. Acad. Sci. Paris, 338 (2004), pp. 951-956, http://dx.doi.org/10.1016/j.crma.2004.04.006 doi:10.1016/j.crma.2004.04.006. · Zbl 1149.76649
[7] J.A. Carrillo, T. Goudon, P. Lafitte, and F. Vecil, {\it Numerical schemes of diffusion asymptotics and moment closures for kinetic equations}, J. Sci. Comput., 36 (2008), pp. 113-149, http://dx.doi.org/10.1007/s10915-007-9181-5 doi:10.1007/s10915-007-9181-5. · Zbl 1203.65015
[8] N. Crouseilles, H. Hivert, and M. Lemou, {\it Numerical schemes for kinetic equations in the anomalous diffusion limit. Part \textupII: The case of degenerate collision frequency}, SIAM J. Sci. Comput., submitted. · Zbl 1515.35032
[9] N. Crouseilles, H. Hivert, and M. Lemou, {\it Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit}, C. R. Math. Acad. Sci. Paris, 353 (2015), pp. 755-760, http://dx.doi.org/10.1016/j.crma.2015.05.003 doi:10.1016/j.crma.2015.05.003. · Zbl 1330.76116
[10] S. De Moor, {\it Fractional diffusion limit for a stochastic kinetic equation}, Stochastic Process. Appl., 124 (2014), pp. 1335-1367, http://dx.doi.org/10.1016/j.spa.2013.11.007 doi:10.1016/j.spa.2013.11.007. · Zbl 1287.60049
[11] S. De Moor, {\it Limites diffusives pour des équations cinétiques stochastiques}, Ph.D. thesis, ENS Rennes, Rennes, France, 2014.
[12] P. Degond, T. Goudon, and F. Poupaud, {\it Diffusion limit for non homogeneous and non-micro-reversible processes}, Indiana Univ. Math. J., 49 (2000), pp. 1175-1198, http://dx.doi.org/10.1512/iumj.2000.49.1936 doi:10.1512/iumj.2000.49.1936. · Zbl 0971.82035
[13] D. del Castillo-Negrete, B. Carreras, and V. Lynch, {\it Non-diffusive transport in plasma turbulence: A fractional diffusion approach}, Phys. Rev. Lett., 94 (2005), 065003, http://dx.doi.org/10.1103/PhysRevLett.94.065003 doi:10.1103/PhysRevLett.94.065003.
[14] M.H. Ernst and R. Brito, {\it Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails}, J. Statist. Phys., 109 (2002), pp. 407-432, http://dx.doi.org/10.1023/A:1020437925931 doi:10.1023/A:1020437925931. · Zbl 1015.82030
[15] M. Hochbruck and A. Ostermann, {\it Exponential integrators}, Acta Numer., 19 (2010), pp. 209-286, http://dx.doi.org/10.1017/S0962492910000048 doi:10.1017/S0962492910000048. · Zbl 1242.65109
[16] S. Jin, {\it Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations}, SIAM J. Sci. Comput., 21 (1999), pp. 441-454, http://dx.doi.org/10.1137/S1064827598334599 doi:10.1137/S1064827598334599. · Zbl 0947.82008
[17] S. Jin and L. Pareschi, {\it Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes}, J. Comput. Phys., 161 (2000), pp. 312-330, http://dx.doi.org/10.1006/jcph.2000.6506 doi:10.1006/jcph.2000.6506. · Zbl 1156.82408
[18] S. Jin, L. Pareschi, and G. Toscani, {\it Uniformly accurate diffusive relaxation schemes for multiscale transport equations}, SIAM J. Numer. Anal., 38 (2000), pp. 913-936, http://dx.doi.org/10.1137/S0036142998347978 doi:10.1137/S0036142998347978. · Zbl 0976.65091
[19] A. Klar, {\it An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit}, SIAM J. Numer. Anal., 35 (1998), pp. 1073-1094, http://dx.doi.org/10.1137/S0036142996305558 doi:10.1137/S0036142996305558. · Zbl 0918.65091
[20] C. Kleiber and S. Kotz, {\it Statistical Size Distributions in Economics and Actuarial Sciences}, John Wiley and Sons, Hoboken, NJ, 2003. · Zbl 1044.62014
[21] E.W. Larsen and J.B. Keller, {\it Asymptotic solution of neutron transport problems for small mean free paths}, J. Math. Phys., 15 (1974), pp. 75-81.
[22] M. Lemou and L. Mieussens, {\it A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit}, SIAM J. Sci. Comput., 31 (2008), pp. 334-368, http://dx.doi.org/10.1137/07069479X doi:10.1137/07069479X. · Zbl 1187.82110
[23] A. Mellet, {\it Fractional diffusion limit for collisional kinetic equations: A moments method}, Indiana Univ. Math. J., 59 (2010), pp. 1333-1360. · Zbl 1294.82032
[24] A. Mellet, S. Mischler, and C. Mouhot, {\it Fractional diffusion limit for collisional kinetic equations}, Arch. Ration. Mech. Anal., 199 (2011), pp. 493-525, http://dx.doi.org/10.1007/s00205-010-0354-2 doi:10.1007/s00205-010-0354-2. · Zbl 1294.82033
[25] D.A. Mendis and M. Rosenberg, {\it Cosmic dusty plasma}, Annu. Rev. Astron. Astrophys., 32 (1994), pp. 419-463, http://dx.doi.org/10.1146/annurev.aa.32.090194.002223 doi:10.1146/annurev.aa.32.090194.002223.
[26] L. Mieussens, {\it On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models}, J. Comput. Phys., 253 (2013), pp. 138-156, http://dx.doi.org/10.1016/j.jcp.2013.07.002 doi:10.1016/j.jcp.2013.07.002. · Zbl 1349.76787
[27] G. Naldi and L. Pareschi, {\it Numerical schemes for kinetic equations in diffusive regimes}, Appl. Math. Lett., 11 (1998), pp. 29-35, http://dx.doi.org/10.1016/S0893-9659(98)00006-8 doi:10.1016/S0893-9659(98)00006-8. · Zbl 1337.65118
[28] D. Summers and R.M. Thorne, {\it The modified plasma dispersion function}, Phys. Fluids B, 3 (1991), pp. 1835-1847, http://dx.doi.org/10.1063/1.859653 doi:10.1063/1.859653.
[29] L. Wang and B. Yan, {\it An Asymptotic-Preserving Scheme for Linear Kinetic Equation with Fractional Diffusion Limit}, preprint, 2015.
[30] E. Wigner, {\it Nuclear Reactor Theory}, AMS, Providence, RI, 1961.
[31] K. Xu, {\it A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method}, J. Comput. Phys., 171 (2001), pp. 289-335, http://dx.doi.org/10.1006/jcph.2001.6790 doi:10.1006/jcph.2001.6790. · Zbl 1058.76056
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