Article Contents
Article Contents

# Numerical schemes for kinetic equation with diffusion limit and anomalous time scale

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The author is supported by ERC starting grant MESOPROBIO

• In this work, we propose numerical schemes for linear kinetic equation, which are able to deal with a diffusion limit and an anomalous time scale of the form ${\varepsilon ^2}\left( {1 + \left| {\ln \left( \varepsilon \right)} \right|} \right)$. When the equilibrium distribution function is a heavy-tailed function, it is known that for an appropriate time scale, the mean-free-path limit leads either to diffusion or fractional diffusion equation, depending on the tail of the equilibrium. This work deals with a critical exponent between these two cases, for which an anomalous time scale must be used to find a standard diffusion limit. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degeneracy $\varepsilon\to0$ makes the kinetic equation stiff. From a numerical point of view, it is necessary to construct schemes able to undertake this stiffness to avoid the increase of computational cost. In this case, it is crucial to capture numerically the effects of the large velocities of the heavy-tailed equilibrium. Moreover, we prove that the convergence towards the diffusion limit happens with two scales, the second being very slow. The schemes we propose are designed to respect this asymptotic behavior. Various numerical tests are performed to illustrate the efficiency of our methods in this context.

Mathematics Subject Classification: Primary: 35B40, 65L04, 41A60; Secondary: 65M22.

 Citation:

• Figure 1.  For $\Delta t = 10^{-2}$, the solutions of (34) at time $T = 0.1$ for different values of $\varepsilon$, when $a_\varepsilon$ is computed with (35), and the solution of the limit scheme (37)

Figure 2.  For $\Delta t = 10^{-2}$, the solutions of (34) at time $T = 0.1$ for different values of $\varepsilon$, when $a_\varepsilon$ is computed with (36), and the solution of the limit scheme (37)

Figure 3.  For $\Delta t = 10^{-2}$, the relative error between the solution of the scheme (34) and the limit scheme (37) at time $T = 0.1$, in function of $\varepsilon$ (log scale)

Figure 6.  For $\Delta t = 10^{-2}$, the solutions of IS, quasi-diff and diff schemes for different values of $\varepsilon$

Figure 7.  For $\Delta t = 10^{-2}$, the error (71) between the solution of IS and quasi-diff scheme in function of $\varepsilon$ (log scale)

Figure 8.  For $\Delta t = 10^{-4}$ and $\varepsilon = 1$, the solutions of the MMS scheme and of the explicit scheme (69)

Figure 9.  The relative consistency error (70) for the MMS scheme (log scale)

Figure 10.  For $\Delta t = 10^{-4}$, the solutions of MMS, quasi-diff and diff schemes for different values of $\varepsilon$

Figure 4.  For $\Delta t = 10^{-2}$ and $\varepsilon = 1$, the solutions of the IS scheme and of the explicit scheme (69)

Figure 5.  The relative consistency error (70) for the IS scheme (log scale)

Figure 11.  For $\Delta t = 10^{-4}$, the error (71) between the solution of MMS and quasi-diff scheme in function of $\varepsilon$ (log scale)

Figure 12.  For $\Delta t = 10^{-2}$ and $\varepsilon = 1$, the solutions of the DS scheme and of the explicit scheme (69)

Figure 13.  The relative consistency error (70) for the DS scheme (log scale)

Figure 14.  For $\Delta t = 10^{-2}$, the solutions of DS, quasi-diff and diff schemes for different values of $\varepsilon$

Figure 15.  For $\Delta t = 10^{-2}$, the error (71) between the solution of DS and quasi-diff scheme in function of $\varepsilon$ (log scale)

Figure 16.  The error (70) as a function of $\varepsilon$. The density $\rho_{reference}$ is the density given by the DS scheme for $\Delta t_{ref} = 5\cdot 10^{-5}$, and $\rho_{\Delta t}$ are the densities given by the DS scheme for different values of $\Delta t$ (log scale)

•  C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. Ⅱ. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. , 46 (1993), 667-753, URL http://dx.doi.org/10.1002/cpa.3160460503. doi: 10.1002/cpa.3160460503. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. Ⅰ. Formal derivations, J. Statist. Phys. , 63 (1991), 323-344, URL http://dx.doi.org/10.1007/BF01026608. doi: 10.1007/BF01026608. C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc. , 284 (1984), 617-649, URL http://dx.doi.org/10.2307/1999099. doi: 10.1090/S0002-9947-1984-0743736-0. N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900, URL http://dx.doi.org/10.3934/krm.2011.4.873. doi: 10.3934/krm.2011.4.873. A. Bensoussan, J. -L. Lions and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci. , 15 (1979), 53-157, URL http://dx.doi.org/10.2977/prims/1195188427. doi: 10.2977/prims/1195188427. A. V. Bobylev, J. A. Carrillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys. , 98 (2000), 743-773, URL http://dx.doi.org/10.1023/A:1018627625800. doi: 10.1023/A:1018627625800. A. V. Bobylev and I. M. Gamba, Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails, J. Stat. Phys. , 124 (2006), 497-516, URL http://dx.doi.org/10.1007/s10955-006-9044-8. doi: 10.1007/s10955-006-9044-8. C. Buet and S. Cordier, Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models, C. R. Math. Acad. Sci. Paris, 338 (2004), 951-956, URL http://dx.doi.org/10.1016/j.crma.2004.04.006. doi: 10.1016/j.crma.2004.04.006. J. A. Carrillo, T. Goudon, P. Lafitte and F. Vecil, Numerical schemes of diffusion asymptotics and moment closures for kinetic equations, J. Sci. Comput. , 36 (2008), 113-149, URL http://dx.doi.org/10.1007/s10915-007-9181-5. doi: 10.1007/s10915-007-9181-5. N. Crouseilles, H. Hivert and M. Lemou, Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit, C. R. Math. Acad. Sci. Paris, 353 (2015), 755-760, URL http://dx.doi.org/10.1016/j.crma.2015.05.003. doi: 10.1016/j.crma.2015.05.003. N. Crouseilles, H. Hivert and M. Lemou, Numerical schemes for kinetic equations in the anomalous diffusion limit. Part Ⅰ: The case of heavy-tailed equilibrium, SIAM J. Sci. Comput., 38 (2016), A737-A764, URL http://dx.doi.org/10.1137/15M1011366. doi: 10.1137/15M1011366. N. Crouseilles, H. Hivert and M. Lemou, Numerical schemes for kinetic equations in the anomalous diffusion limit. Part Ⅱ: Degenerate collision frequency, SIAM J. Sci. Comput. , 38 (2016), A2464-A2491, URL http://dx.doi.org/10.1137/15M1053190. doi: 10.1137/15M1053190. S. De Moor, Fractional diffusion limit for a stochastic kinetic equation, Stochastic Process. Appl. , 124 (2014), 1335-1367, URL http://dx.doi.org/10.1016/j.spa.2013.11.007. doi: 10.1016/j.spa.2013.11.007. S. De Moor, Limites Diffusives Pour des Équations Cinétiques Stochastiques, PhD thesis, ENS Rennes, Rennes, France, 2014. P. Degond , T. Goudon  and  F. Poupaud , Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000) , 1175-1198. D. del Castillo-Negrete, B. Carreras and V. Lynch, Non-diffusive transport in plasma turbulence: A fractional diffusion approach, Phys. Rev. Lett. , 94 (2005), 065003, URL http://link.aps.org/doi/10.1103/PhysRevLett.94.065003. M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys. , 109 (2002), 407-432, URL http://dx.doi.org/10.1023/A:1020437925931, Special issue dedicated to J. Robert Dorfman on the occasion of his sixty-fifth birthday. doi: 10.1023/A:1020437925931. H. Hivert, Mathematical and Numerical Study of Some Kinetic Models and of Their Asymptotics: Diffusion and Anomalous Diffusion Limits, Theses, Université Rennes 1,2016, URL https://tel.archives-ouvertes.fr/tel-01393275. H. Hivert, A relaxed micro-macro scheme for kinetic equation with fractional diffusion limit and non-local collision operator, in preparation. S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput. , 21 (1999), 441-454, URL http://dx.doi.org/10.1137/S1064827598334599. doi: 10.1137/S1064827598334599. S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comput. Phys. , 161 (2000), 312-330, URL http://dx.doi.org/10.1006/jcph.2000.6506. doi: 10.1006/jcph.2000.6506. S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal. , 38 (2000), 913-936, URL http://dx.doi.org/10.1137/S0036142998347978. doi: 10.1137/S0036142998347978. A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal. , 35 (1998), 1073-1094, URL http://dx.doi.org/10.1137/S0036142996305558. doi: 10.1137/S0036142996305558. A. Klar, An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal. , 36 (1999), 1507-1527, URL http://dx.doi.org/10.1137/S0036142997321765. doi: 10.1137/S0036142997321765. C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2003, URL http://dx.doi.org/10.1002/0471457175. M. Lemou, Relaxed micro-macro schemes for kinetic equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 455-460, URL http://dx.doi.org/10.1016/j.crma.2010.02.017. doi: 10.1016/j.crma.2010.02.017. M. Lemou and F. Méhats, Micro-macro schemes for kinetic equations including boundary layers, SIAM J. Sci. Comput. , 34 (2012), B734-B760, URL http://dx.doi.org/10.1137/120865513. doi: 10.1137/120865513. M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput. , 31 (2008), 334-368, URL http://dx.doi.org/10.1137/07069479X. doi: 10.1137/07069479X. A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal. , 199 (2011), 493-525, URL http://dx.doi.org/10.1007/s00205-010-0354-2. doi: 10.1007/s00205-010-0354-2. D. A. Mendis  and  M. Rosenberg , Cosmic dusty plasma, Annu. Rev. Astron. Astrophys., 32 (1994) , 419-463.  doi: 10.1146/annurev.aa.32.090194.002223. D. Summers and R. M. Thorne, The modified plasma dispersion function, Fluids B, 3 (1991), 1835-1847, URL http://scitation.aip.org/content/aip/journal/pofb/3/8/10.1063/1.859653. doi: 10.1063/1.859653. L. Wang and B. Yan, An asymptotic-preserving scheme for kinetic equation with anisotropic scattering: heavy-tail equilibrium and degenerate collision frequency, preprint. L. Wang and B. Yan, An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit, J. Comput. Phys. , 312 (2016), 157-174, URL http://dx.doi.org/10.1016/j.jcp.2016.02.034. doi: 10.1016/j.jcp.2016.02.034. E. Wigner, Nuclear Reactor Theory, AMS, Providence, RI, 1961.

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