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September  2014, 7(3): 463-476. doi: 10.3934/krm.2014.7.463

Numerical simulations of degenerate transport problems

1. 

Centre de Mathématiques et de leurs Applications, CMLA UMR 8536, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan, France

2. 

Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1 - 27100 Pavia

Received  September 2013 Revised  February 2014 Published  July 2014

We consider in this article the monokinetic linear Boltzmann equation in two space dimensions with degenerate cross section and produce, by means of a finite-volume method, numerical simulations of the large-time asymptotics of the solution.
    The numerical computations are performed in the $2Dx-1Dv$ phase space on Cartesian grids and deal with both cross sections satisfying the geometrical condition and cross sections that do not satisfy it.
    The numerical simulations confirm the theoretical results on the long-time behaviour of degenerate kinetic equations for cross sections satisfying the geometrical condition. Moreover, they suggest that, for general non-trivial degenerate cross sections whose support contains a ball, the theoretical upper bound of order $t^{-1/2}$ for the time decay rate (in $L^2$-sense) can actually be reached.
Citation: Florian De Vuyst, Francesco Salvarani. Numerical simulations of degenerate transport problems. Kinetic and Related Models, 2014, 7 (3) : 463-476. doi: 10.3934/krm.2014.7.463
References:
[1]

E. Bernard and F. Salvarani, Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model, J. Stat. Phys., 153 (2013), 363-375. doi: 10.1007/s10955-013-0825-6.

[2]

E. Bernard and F. Salvarani, On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, J. Funct. Anal., 265 (2013), 1934-1954. doi: 10.1016/j.jfa.2013.06.012.

[3]

E. Bernard and F. Salvarani, On the convergence to equilibrium for degenerate transport problems, Arch. Ration. Mech. Anal., 208 (2013), 977-984. doi: 10.1007/s00205-012-0608-2.

[4]

S. Brull and L. Mieussens, Local discrete velocity grids for deterministic rarefied flow simulations, Journal of Computational Physics, 266 (2014), 22-46. doi: 10.1016/j.jcp.2014.01.050.

[5]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.

[6]

F. De Vuyst and F. Salvarani, GPU-accelerated numerical simulations of the Knudsen gas on time-dependent domains, Comput. Phys. Comm., 184 (2013), 532-536. doi: 10.1016/j.cpc.2012.10.004.

[7]

L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Sci. Math., 133 (2009), 848-858. doi: 10.1016/j.bulsci.2008.09.001.

[8]

S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156. doi: 10.1093/qjmam/4.2.129.

[9]

D. Han-Kwan and M. Léautaud, Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium, arXiv:1401.8227, (2014).

[10]

A. Kurganov and J. Rauch, The order of accuracy of quadrature formulae for periodic functions, in Advances in Phase Space Analysis of Partial Differential Equations, Progr. Nonlinear Differential Equations Appl., 78, Birkhäuser Boston, Inc., Boston, MA, 2009, 155-159. doi: 10.1007/978-0-8176-4861-9_9.

[11]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, John Wiley and Sons, Inc., New York, NY, 1984.

[12]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory, Series on Advances in Mathematics for Applied Sciences, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.

[13]

L. Neumann and C. Mouhot, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

[14]

F. Salvarani, On the linear Boltzmann equation in evolutionary domains with absorbing boundary, J. Phys. A: Math. Theor., 46 (2013), 355501, 15 pp. doi: 10.1088/1751-8113/46/35/355501.

[15]

G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc., S2-20 (1922), 196. doi: 10.1112/plms/s2-20.1.196.

[16]

S. Ukai, N. Point and H. Ghidouche, Sur la solution globale du problème mixte de l'équation de Boltzmann nonlinéaire, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 57 (1978), 203-229.

show all references

References:
[1]

E. Bernard and F. Salvarani, Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model, J. Stat. Phys., 153 (2013), 363-375. doi: 10.1007/s10955-013-0825-6.

[2]

E. Bernard and F. Salvarani, On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, J. Funct. Anal., 265 (2013), 1934-1954. doi: 10.1016/j.jfa.2013.06.012.

[3]

E. Bernard and F. Salvarani, On the convergence to equilibrium for degenerate transport problems, Arch. Ration. Mech. Anal., 208 (2013), 977-984. doi: 10.1007/s00205-012-0608-2.

[4]

S. Brull and L. Mieussens, Local discrete velocity grids for deterministic rarefied flow simulations, Journal of Computational Physics, 266 (2014), 22-46. doi: 10.1016/j.jcp.2014.01.050.

[5]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.

[6]

F. De Vuyst and F. Salvarani, GPU-accelerated numerical simulations of the Knudsen gas on time-dependent domains, Comput. Phys. Comm., 184 (2013), 532-536. doi: 10.1016/j.cpc.2012.10.004.

[7]

L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Sci. Math., 133 (2009), 848-858. doi: 10.1016/j.bulsci.2008.09.001.

[8]

S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156. doi: 10.1093/qjmam/4.2.129.

[9]

D. Han-Kwan and M. Léautaud, Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium, arXiv:1401.8227, (2014).

[10]

A. Kurganov and J. Rauch, The order of accuracy of quadrature formulae for periodic functions, in Advances in Phase Space Analysis of Partial Differential Equations, Progr. Nonlinear Differential Equations Appl., 78, Birkhäuser Boston, Inc., Boston, MA, 2009, 155-159. doi: 10.1007/978-0-8176-4861-9_9.

[11]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, John Wiley and Sons, Inc., New York, NY, 1984.

[12]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory, Series on Advances in Mathematics for Applied Sciences, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.

[13]

L. Neumann and C. Mouhot, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

[14]

F. Salvarani, On the linear Boltzmann equation in evolutionary domains with absorbing boundary, J. Phys. A: Math. Theor., 46 (2013), 355501, 15 pp. doi: 10.1088/1751-8113/46/35/355501.

[15]

G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc., S2-20 (1922), 196. doi: 10.1112/plms/s2-20.1.196.

[16]

S. Ukai, N. Point and H. Ghidouche, Sur la solution globale du problème mixte de l'équation de Boltzmann nonlinéaire, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 57 (1978), 203-229.

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