American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space
  • KRM Home
  • This Issue
  • Next Article
    Long time strong convergence to Bose-Einstein distribution for low temperature
August  2018, 11(4): 735-756. doi: 10.3934/krm.2018030

An asymptotic preserving scheme for kinetic models with singular limit

Alina Chertock 1,, , Changhui Tan 2, and  Bokai Yan 3,

1. 

Department of Mathematics, North Carolina State University, Campus Box 8205, Raleigh NC 27695, USA
2. 

Department of Mathematics, Rice University, 6100 Main St., Houston, TX 77005, USA
3. 

Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA

* Corresponding author: Alina Chertock

Received  June 2017 Revised  November 2017 Published  April 2018

We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors, and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.

Keywords: Kinetic models, singular limit, asymptotic-preserving schemes, velocity scaling, aggregation equation, flocking phenomena.
Mathematics Subject Classification: Primary: 82C40, 35L65; Secondary: 65M08.
Citation: 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
References:
[1]

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.  doi: 10.1023/A:1018627625800.  Google Scholar

[2]

M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: A formal approach, Math. Methods Appl. Sci., 28 (2005), 1757-1779.  doi: 10.1002/mma.638.  Google Scholar

[3]

J. A. Carrillo, Y.-P. Choi, E. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26 (2016), 185-206.  doi: 10.1142/S0218202516500068.  Google Scholar

[4]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[5]

N. Crouseilles, H. Hilvert 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.  doi: 10.1137/15M1011366.  Google Scholar

[6]

N. Crouseilles, H. Hilvert 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.  doi: 10.1137/15M1053190.  Google Scholar

[7]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[8]

T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Ration. Mech. Anal., 228 (2018), 1-37. doi: 10.1007/s00205-017-1184-2.  Google Scholar

[9]

R. C. Fetecau and W. Sun, First-order aggregation models and zero inertia limits, J. Differential Equations, 259 (2015), 6774-6802.  doi: 10.1016/j.jde.2015.08.018.  Google Scholar

[10]

R. C. Fetecau, W. Sun and C. Tan, First-order aggregation models with alignment, Phys. D, 325 (2016), 146-163.  doi: 10.1016/j.physd.2016.03.011.  Google Scholar

[11]

F. Filbet and T. Rey, A rescaling velocity method for dissipative kinetic equations. Applications to granular media, J. Comput. Phys., 248 (2013), 177-199.  doi: 10.1016/j.jcp.2013.04.023.  Google Scholar

[12]

F. Filbet and G. Russo, A rescaling velocity method for kinetic equations: The homogeneous case, Modelling and numerics of kinetic dissipative systems, 191-202, Nova Sci. Publ., Hauppauge, NY, 2006.  Google Scholar

[13]

T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows, J. Comput. Phys., 246 (2013), 145-164.  doi: 10.1016/j.jcp.2013.03.038.  Google Scholar

[14]

T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows with variable fluid density, Internat. J. Numer. Methods Fluids, 75 (2014), 81-102.  doi: 10.1002/fld.3885.  Google Scholar

[15]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[16]

P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672.  doi: 10.1016/S0294-1449(00)00118-9.  Google Scholar

[17]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[18]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.), 3 (2012), 177-216.   Google Scholar

[19]

A. Kiselev and C. Tan, Global regularity for 1D Eulerian dynamics with singular interaction forces, preprint, arXiv: 1707.07296. Google Scholar

[20]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[21]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089-1152.  doi: 10.1142/S0218202517400103.  Google Scholar

[22]

T. Rey and C. Tan, An exact rescaling velocity method for some kinetic flocking models, SIAM J. Numer. Anal., 54 (2016), 641-664.  doi: 10.1137/140993430.  Google Scholar

[23]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Comput. Graph (ACM), 21 (1987), 25-34.  doi: 10.1145/37401.37406.  Google Scholar

[24]

E. Tadmor and C. Tadmor, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401, 22pp..  doi: 10.1098/rsta.2013.0401.  Google Scholar

[25]

C. Tan, A discontinuous Galerkin method on kinetic flocking models, Math. Models Methods Appl. Sci., 27 (2017), 1199-1221.  doi: 10.1142/S0218202517400139.  Google Scholar

[26]

C. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[27]

L. Wang and B. Yan, An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit, J. Comput. Phys., 312 (2016), 157-174.  doi: 10.1016/j.jcp.2016.02.034.  Google Scholar

[28]

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

show all references

References:
[1]

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.  doi: 10.1023/A:1018627625800.  Google Scholar

[2]

M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: A formal approach, Math. Methods Appl. Sci., 28 (2005), 1757-1779.  doi: 10.1002/mma.638.  Google Scholar

[3]

J. A. Carrillo, Y.-P. Choi, E. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26 (2016), 185-206.  doi: 10.1142/S0218202516500068.  Google Scholar

[4]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[5]

N. Crouseilles, H. Hilvert 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.  doi: 10.1137/15M1011366.  Google Scholar

[6]

N. Crouseilles, H. Hilvert 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.  doi: 10.1137/15M1053190.  Google Scholar

[7]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[8]

T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Ration. Mech. Anal., 228 (2018), 1-37. doi: 10.1007/s00205-017-1184-2.  Google Scholar

[9]

R. C. Fetecau and W. Sun, First-order aggregation models and zero inertia limits, J. Differential Equations, 259 (2015), 6774-6802.  doi: 10.1016/j.jde.2015.08.018.  Google Scholar

[10]

R. C. Fetecau, W. Sun and C. Tan, First-order aggregation models with alignment, Phys. D, 325 (2016), 146-163.  doi: 10.1016/j.physd.2016.03.011.  Google Scholar

[11]

F. Filbet and T. Rey, A rescaling velocity method for dissipative kinetic equations. Applications to granular media, J. Comput. Phys., 248 (2013), 177-199.  doi: 10.1016/j.jcp.2013.04.023.  Google Scholar

[12]

F. Filbet and G. Russo, A rescaling velocity method for kinetic equations: The homogeneous case, Modelling and numerics of kinetic dissipative systems, 191-202, Nova Sci. Publ., Hauppauge, NY, 2006.  Google Scholar

[13]

T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows, J. Comput. Phys., 246 (2013), 145-164.  doi: 10.1016/j.jcp.2013.03.038.  Google Scholar

[14]

T. Goudon, S. Jin, J.-G. Liu and B. Yan, Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows with variable fluid density, Internat. J. Numer. Methods Fluids, 75 (2014), 81-102.  doi: 10.1002/fld.3885.  Google Scholar

[15]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[16]

P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672.  doi: 10.1016/S0294-1449(00)00118-9.  Google Scholar

[17]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[18]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.), 3 (2012), 177-216.   Google Scholar

[19]

A. Kiselev and C. Tan, Global regularity for 1D Eulerian dynamics with singular interaction forces, preprint, arXiv: 1707.07296. Google Scholar

[20]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[21]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089-1152.  doi: 10.1142/S0218202517400103.  Google Scholar

[22]

T. Rey and C. Tan, An exact rescaling velocity method for some kinetic flocking models, SIAM J. Numer. Anal., 54 (2016), 641-664.  doi: 10.1137/140993430.  Google Scholar

[23]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Comput. Graph (ACM), 21 (1987), 25-34.  doi: 10.1145/37401.37406.  Google Scholar

[24]

E. Tadmor and C. Tadmor, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401, 22pp..  doi: 10.1098/rsta.2013.0401.  Google Scholar

[25]

C. Tan, A discontinuous Galerkin method on kinetic flocking models, Math. Models Methods Appl. Sci., 27 (2017), 1199-1221.  doi: 10.1142/S0218202517400139.  Google Scholar

[26]

C. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[27]

L. Wang and B. Yan, An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit, J. Comput. Phys., 312 (2016), 157-174.  doi: 10.1016/j.jcp.2016.02.034.  Google Scholar

[28]

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

Figure 1.  AP scheme under transformation
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Example 1: The test on assumption 49. From left to right: the time evolution of $\max\limits_x |\nabla _x u_\varepsilon |$, $\max\limits_x \frac{|\nabla _x \rho_\varepsilon |}{\rho_\varepsilon }$ and $\max\limits_x \frac{|\nabla _x P_\varepsilon |}{\rho_\varepsilon }$ for different values of $\varepsilon $. The lines for $\varepsilon = 10^{-3}$ and $\varepsilon = 10^{-4}$ are almost overlapped.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Example 2: Top left: the time evolution of $\rho_1(x)$ solved from the original system (1) (blue solid lines) and the rescaled system (41) (red dashed lines). Top right: the time evolution of $u_1(x)$ solved from the original system (1) (blue solid lines) and the rescaled system (41) (red dashed lines). Bottom left: the distribution $f_1(x, v)$ at time $t = 0.7$ solved from the original system (1). Bottom right: the distribution $g_1(x, \xi)$ at time $t = 0.7$ solved from the rescaled system (41).
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Example 3: The density $\rho_\varepsilon (x)$ (left) and the macroscopic velocity $u_\varepsilon (x)$ (right) at time $t = 1$ computed by the scheme (42) with different $\varepsilon $'s are present, as well as that of the limiting system (6). The lines corresponding to $\varepsilon = 10^{-3}$ almost overlap with the lines of limiting system.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Example 4: Time snapshots of the solution to the aggregation system. From left to right: the distribution $g_\varepsilon (x, \xi)$, the density $\rho_\varepsilon (x)$, the momentum $\rho_\varepsilon (x)u_\varepsilon (x)$ and the scaling factor $\omega_\varepsilon (x)$. In this test $\varepsilon = 1$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Example 4: Time snapshots of the solution to the aggregation system. From left to right: the distribution $g_\varepsilon (x, \xi)$, the density $\rho_\varepsilon (x)$, the momentum $\rho_\varepsilon (x)u_\varepsilon (x)$ and the scaling factor $\omega_\varepsilon (x)$. In this test $\varepsilon = 10^{-4}$. The stationary solution $\rho$ and $\rho u$ of the limiting system (6) is illustrated by red dashed lines in the last row.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[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]

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
[5]

Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573
[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]

Le Li, Lihong Huang, Jianhong Wu. Flocking and invariance of velocity angles. Mathematical Biosciences & Engineering, 2016, 13 (2) : 369-380. doi: 10.3934/mbe.2015007
[8]

Nicolas Crouseilles, Giacomo Dimarco, Mohammed Lemou. Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinetic & Related Models, 2017, 10 (3) : 643-668. doi: 10.3934/krm.2017026
[9]

Kazuo Aoki, Ansgar Jüngel, Peter A. Markowich. Small velocity and finite temperature variations in kinetic relaxation models. Kinetic & Related Models, 2010, 3 (1) : 1-15. doi: 10.3934/krm.2010.3.1
[10]

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
[11]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35
[12]

Giada Basile, Tomasz Komorowski, Stefano Olla. Diffusion limit for a kinetic equation with a thermostatted interface. Kinetic & Related Models, 2019, 12 (5) : 1185-1196. doi: 10.3934/krm.2019045
[13]

Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170
[14]

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
[15]

Vincent Calvez, Gaël Raoul, Christian Schmeiser. Confinement by biased velocity jumps: Aggregation of escherichia coli. Kinetic & Related Models, 2015, 8 (4) : 651-666. doi: 10.3934/krm.2015.8.651
[16]

Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415
[17]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for the Ibragimov-Shabat equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 661-673. doi: 10.3934/dcdss.2016020
[18]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146
[19]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020190
[20]

Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic & Related Models, 2017, 10 (3) : 541-551. doi: 10.3934/krm.2017021

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (74)
  • HTML views (179)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences