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Long time strong convergence to Bose-Einstein distribution for low temperature
An asymptotic preserving scheme for kinetic models with singular limit
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[19] |
A. Kiselev and C. Tan, Global regularity for 1D Eulerian dynamics with singular interaction forces, preprint, arXiv: 1707.07296. |
[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. |
[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. |
[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. |
[23] |
C. W. Reynolds,
Flocks, herds and schools: A distributed behavioral model, Comput. Graph (ACM), 21 (1987), 25-34.
doi: 10.1145/37401.37406. |
[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. |
[25] |
C. Tan,
A discontinuous Galerkin method on kinetic flocking models, Math. Models Methods Appl. Sci., 27 (2017), 1199-1221.
doi: 10.1142/S0218202517400139. |
[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. |
[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. |
[28] |
L. Wang and B. Yan, An asymptotic-preserving scheme for kinetic equation with anisotropic scattering: heavy tail equilibrium and degenerate collision frequency, preprint. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[19] |
A. Kiselev and C. Tan, Global regularity for 1D Eulerian dynamics with singular interaction forces, preprint, arXiv: 1707.07296. |
[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. |
[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. |
[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. |
[23] |
C. W. Reynolds,
Flocks, herds and schools: A distributed behavioral model, Comput. Graph (ACM), 21 (1987), 25-34.
doi: 10.1145/37401.37406. |
[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. |
[25] |
C. Tan,
A discontinuous Galerkin method on kinetic flocking models, Math. Models Methods Appl. Sci., 27 (2017), 1199-1221.
doi: 10.1142/S0218202517400139. |
[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. |
[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. |
[28] |
L. Wang and B. Yan, An asymptotic-preserving scheme for kinetic equation with anisotropic scattering: heavy tail equilibrium and degenerate collision frequency, preprint. |






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