April  2018, 11(2): 409-439. doi: 10.3934/krm.2018019

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

ENS Lyon, UMPA UMR 5669 CNRS, and Inria Rhône-Alpes, projet NUMED, 46, allée d'Italie, 69364 Lyon Cedex 07, France

* Corresponding author

Received  October 2016 Revised  May 2017 Published  January 2018

Fund Project: 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.

Citation: Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019
References:
[1]

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. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

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. Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

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. Google Scholar

[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. Google Scholar

[9]

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. Google Scholar

[10]

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. Google Scholar

[11]

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. Google Scholar

[12]

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. Google Scholar

[13]

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. Google Scholar

[14]

S. De Moor, Limites Diffusives Pour des Équations Cinétiques Stochastiques, PhD thesis, ENS Rennes, Rennes, France, 2014.Google Scholar

[15]

P. DegondT. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198. Google Scholar

[16]

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.Google Scholar

[17]

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. Google Scholar

[18]

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.Google Scholar

[19]

H. Hivert, A relaxed micro-macro scheme for kinetic equation with fractional diffusion limit and non-local collision operator, in preparation.Google Scholar

[20]

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. Google Scholar

[21]

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. Google Scholar

[22]

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. Google Scholar

[23]

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. Google Scholar

[24]

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. Google Scholar

[25]

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. Google Scholar

[26]

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. Google Scholar

[27]

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. Google Scholar

[28]

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. Google Scholar

[29]

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. Google Scholar

[30]

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. Google Scholar

[31]

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. Google Scholar

[32]

L. Wang and B. Yan, An asymptotic-preserving scheme for kinetic equation with anisotropic scattering: heavy-tail equilibrium and degenerate collision frequency, preprint.Google Scholar

[33]

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. Google Scholar

[34]

E. Wigner, Nuclear Reactor Theory, AMS, Providence, RI, 1961.Google Scholar

show all references

References:
[1]

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. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

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. Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

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. Google Scholar

[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. Google Scholar

[9]

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. Google Scholar

[10]

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. Google Scholar

[11]

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. Google Scholar

[12]

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. Google Scholar

[13]

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. Google Scholar

[14]

S. De Moor, Limites Diffusives Pour des Équations Cinétiques Stochastiques, PhD thesis, ENS Rennes, Rennes, France, 2014.Google Scholar

[15]

P. DegondT. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198. Google Scholar

[16]

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.Google Scholar

[17]

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. Google Scholar

[18]

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.Google Scholar

[19]

H. Hivert, A relaxed micro-macro scheme for kinetic equation with fractional diffusion limit and non-local collision operator, in preparation.Google Scholar

[20]

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. Google Scholar

[21]

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. Google Scholar

[22]

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. Google Scholar

[23]

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. Google Scholar

[24]

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. Google Scholar

[25]

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. Google Scholar

[26]

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. Google Scholar

[27]

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. Google Scholar

[28]

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. Google Scholar

[29]

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. Google Scholar

[30]

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. Google Scholar

[31]

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. Google Scholar

[32]

L. Wang and B. Yan, An asymptotic-preserving scheme for kinetic equation with anisotropic scattering: heavy-tail equilibrium and degenerate collision frequency, preprint.Google Scholar

[33]

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. Google Scholar

[34]

E. Wigner, Nuclear Reactor Theory, AMS, Providence, RI, 1961.Google Scholar

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)
[1]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[2]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707

[3]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[4]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[5]

Nicolas Crouseilles, Mohammed Lemou. An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits. Kinetic & Related Models, 2011, 4 (2) : 441-477. doi: 10.3934/krm.2011.4.441

[6]

Stéphane Brull, Pierre Degond, Fabrice Deluzet, Alexandre Mouton. Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model. Kinetic & Related Models, 2011, 4 (4) : 991-1023. doi: 10.3934/krm.2011.4.991

[7]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644

[8]

Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4929-4943. doi: 10.3934/dcds.2019201

[9]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391

[10]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[11]

Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 781-795. doi: 10.3934/dcdss.2020044

[12]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

[13]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353

[14]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[15]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[16]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735

[17]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[18]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663

[19]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011

[20]

Rhudaina Z. Mohammad, Karel Švadlenka. Multiphase volume-preserving interface motions via localized signed distance vector scheme. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 969-988. doi: 10.3934/dcdss.2015.8.969

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (53)
  • HTML views (296)
  • Cited by (0)

Other articles
by authors

[Back to Top]