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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 & 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.  doi: 10.1007/s10955-013-0825-6.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

K. M. Case and P. F. Zweifel, Linear Transport Theory,, Addison-Wesley Publishing Co., (1967).   Google Scholar

[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.  doi: 10.1016/j.cpc.2012.10.004.  Google Scholar

[7]

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

[8]

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

[9]

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

[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, (2009), 155.  doi: 10.1007/978-0-8176-4861-9_9.  Google Scholar

[11]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport,, John Wiley and Sons, (1984).   Google Scholar

[12]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory,, Series on Advances in Mathematics for Applied Sciences, (1997).  doi: 10.1142/9789812819833.  Google Scholar

[13]

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

[14]

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

[15]

G. I. Taylor, Diffusion by continuous movements,, Proc. London Math. Soc., S2-20 (1922), 2.  doi: 10.1112/plms/s2-20.1.196.  Google Scholar

[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.   Google Scholar

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.  doi: 10.1007/s10955-013-0825-6.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

K. M. Case and P. F. Zweifel, Linear Transport Theory,, Addison-Wesley Publishing Co., (1967).   Google Scholar

[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.  doi: 10.1016/j.cpc.2012.10.004.  Google Scholar

[7]

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

[8]

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

[9]

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

[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, (2009), 155.  doi: 10.1007/978-0-8176-4861-9_9.  Google Scholar

[11]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport,, John Wiley and Sons, (1984).   Google Scholar

[12]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory,, Series on Advances in Mathematics for Applied Sciences, (1997).  doi: 10.1142/9789812819833.  Google Scholar

[13]

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

[14]

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

[15]

G. I. Taylor, Diffusion by continuous movements,, Proc. London Math. Soc., S2-20 (1922), 2.  doi: 10.1112/plms/s2-20.1.196.  Google Scholar

[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.   Google Scholar

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