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Nonlinear stability of a Vlasov equation for magnetic plasmas
The moment guided Monte Carlo method for the Boltzmann equation
1. | Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, 118, route de Narbonne 31062 TOULOUSE Cedex, France |
References:
[1] |
H. Babovsky, On a simulation scheme for the Boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233.
doi: 10.1002/mma.1670080114. |
[2] |
G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows," Oxford Engineering Science Series, 42, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. |
[3] |
C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation," Institute of Physics (IOP), Series in Plasma Physics, 2004. |
[4] |
J.-F. Bourgat, P. Le Tallec and M. D. Tidriri, Coupling Boltzmann and Navier-Stokes equations by friction, J. Comput. Phys., 127 (1996), 227-245.
doi: 10.1006/jcph.1996.0172. |
[5] |
J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation, J. Comput. Phys., 228 (2009), 460-475. |
[6] |
R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods, in "Acta Numerica, 1998," Acta Numer., 7, Cambridge Univ. Press, Cambridge, (1998), 1-49.
doi: 10.1017/S0962492900002804. |
[7] |
R. E. Caflisch and L. Pareschi, Towards an hybrid method for rarefied gas dynamics, in "Transport in Transition Regimes" (Minneapolis, MN, 2000), IMA Vol. App. Math., 135, Springer, New York, (2004), 57-73.
doi: 10.1007/978-1-4613-0017-5_3. |
[8] |
C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[9] |
A. Crestetto, N. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, to appear in KRM, 2012. |
[10] |
N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the gas-dynamics Boltzmann BGK equation, J. Comput. Phys., 199 (2004), 776-808.
doi: 10.1016/j.jcp.2004.03.007. |
[11] |
P. Degond, G. Dimarco and L. Pareschi, The moment-guided Monte Carlo method, Int. J. Num. Meth. Fluids, 67 (2011), 189-213.
doi: 10.1002/fld.2345. |
[12] |
P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Model. Simul., 5 (2006), 940-979.
doi: 10.1137/060651574. |
[13] |
P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct Monte Carlo, J. Comp. Phys., 231 (2012), 2414-2437.
doi: 10.1016/j.jcp.2011.11.030. |
[14] |
P. Degond, G. Dimarco and L. Mieussens, A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comp. Phys., 229 (2010), 4907-4933.
doi: 10.1016/j.jcp.2010.03.009. |
[15] |
G. Dimarco and L. Pareschi, Hybrid multiscale methods. I. Hyperbolic relaxation problems, Comm. Math. Sci., 1 (2006), 155-177. |
[16] |
G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations, SIAM J. Sci. Comput., 32 (2010), 603-634.
doi: 10.1137/080730585. |
[17] |
G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM J. Num. Anal., 49 (2011), 2057-2077.
doi: 10.1137/100811052. |
[18] |
F. Filbet and S. Jin, A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625-7648.
doi: 10.1016/j.jcp.2010.06.017. |
[19] |
F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46 (2011), 204-224.
doi: 10.1007/s10915-010-9394-x. |
[20] |
W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. |
[21] |
D. B. Hash and H. A. Hassan, Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods, J. Thermophys. Heat Transf., 10 (1996), 242-249. |
[22] |
T. Homolle and N. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comp. Phys., 226 (2007), 2341-2358.
doi: 10.1016/j.jcp.2007.07.006. |
[23] |
T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701. |
[24] |
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454.
doi: 10.1137/S1064827598334599. |
[25] |
S. Jin and Z. P. Xin, Relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276.
doi: 10.1002/cpa.3160480303. |
[26] |
P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes, J. Comput. Phys., 136 (1997), 51-67.
doi: 10.1006/jcph.1997.5729. |
[27] |
R. J. LeVeque, "Numerical Methods for Conservation Laws," Lectures in Mathematics, Birkhäuser Verlag, Basel, 1990. |
[28] |
S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer, 2004. |
[29] |
K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), 2042-2049. |
[30] |
D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34 (1980), 231-244. |
[31] |
S. Tiwari, Coupling of the Boltzmann and Euler equations with automatic domain decomposition, J. Comput. Phys., 144 (1998), 710-726.
doi: 10.1006/jcph.1998.6011. |
[32] |
S. Tiwari, A. Klar and S. Hardt, A particle-particle hybrid method for kinetic and continuum equations, J. Comput. Phys., 228 (2009), 7109-7124.
doi: 10.1016/j.jcp.2009.06.019. |
show all references
References:
[1] |
H. Babovsky, On a simulation scheme for the Boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233.
doi: 10.1002/mma.1670080114. |
[2] |
G. A. Bird, "Molecular Gas Dynamics and Direct Simulation of Gas Flows," Oxford Engineering Science Series, 42, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. |
[3] |
C. K. Birsdall and A. B. Langdon, "Plasma Physics Via Computer Simulation," Institute of Physics (IOP), Series in Plasma Physics, 2004. |
[4] |
J.-F. Bourgat, P. Le Tallec and M. D. Tidriri, Coupling Boltzmann and Navier-Stokes equations by friction, J. Comput. Phys., 127 (1996), 227-245.
doi: 10.1006/jcph.1996.0172. |
[5] |
J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation, J. Comput. Phys., 228 (2009), 460-475. |
[6] |
R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods, in "Acta Numerica, 1998," Acta Numer., 7, Cambridge Univ. Press, Cambridge, (1998), 1-49.
doi: 10.1017/S0962492900002804. |
[7] |
R. E. Caflisch and L. Pareschi, Towards an hybrid method for rarefied gas dynamics, in "Transport in Transition Regimes" (Minneapolis, MN, 2000), IMA Vol. App. Math., 135, Springer, New York, (2004), 57-73.
doi: 10.1007/978-1-4613-0017-5_3. |
[8] |
C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[9] |
A. Crestetto, N. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, to appear in KRM, 2012. |
[10] |
N. Crouseilles, P. Degond and M. Lemou, A hybrid kinetic-fluid model for solving the gas-dynamics Boltzmann BGK equation, J. Comput. Phys., 199 (2004), 776-808.
doi: 10.1016/j.jcp.2004.03.007. |
[11] |
P. Degond, G. Dimarco and L. Pareschi, The moment-guided Monte Carlo method, Int. J. Num. Meth. Fluids, 67 (2011), 189-213.
doi: 10.1002/fld.2345. |
[12] |
P. Degond, J.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Model. Simul., 5 (2006), 940-979.
doi: 10.1137/060651574. |
[13] |
P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct Monte Carlo, J. Comp. Phys., 231 (2012), 2414-2437.
doi: 10.1016/j.jcp.2011.11.030. |
[14] |
P. Degond, G. Dimarco and L. Mieussens, A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comp. Phys., 229 (2010), 4907-4933.
doi: 10.1016/j.jcp.2010.03.009. |
[15] |
G. Dimarco and L. Pareschi, Hybrid multiscale methods. I. Hyperbolic relaxation problems, Comm. Math. Sci., 1 (2006), 155-177. |
[16] |
G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations, SIAM J. Sci. Comput., 32 (2010), 603-634.
doi: 10.1137/080730585. |
[17] |
G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM J. Num. Anal., 49 (2011), 2057-2077.
doi: 10.1137/100811052. |
[18] |
F. Filbet and S. Jin, A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625-7648.
doi: 10.1016/j.jcp.2010.06.017. |
[19] |
F. Filbet and S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46 (2011), 204-224.
doi: 10.1007/s10915-010-9394-x. |
[20] |
W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. |
[21] |
D. B. Hash and H. A. Hassan, Assessment of schemes for coupling Monte Carlo and Navier-Stokes solution methods, J. Thermophys. Heat Transf., 10 (1996), 242-249. |
[22] |
T. Homolle and N. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comp. Phys., 226 (2007), 2341-2358.
doi: 10.1016/j.jcp.2007.07.006. |
[23] |
T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701. |
[24] |
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454.
doi: 10.1137/S1064827598334599. |
[25] |
S. Jin and Z. P. Xin, Relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-276.
doi: 10.1002/cpa.3160480303. |
[26] |
P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes, J. Comput. Phys., 136 (1997), 51-67.
doi: 10.1006/jcph.1997.5729. |
[27] |
R. J. LeVeque, "Numerical Methods for Conservation Laws," Lectures in Mathematics, Birkhäuser Verlag, Basel, 1990. |
[28] |
S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer, 2004. |
[29] |
K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), 2042-2049. |
[30] |
D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34 (1980), 231-244. |
[31] |
S. Tiwari, Coupling of the Boltzmann and Euler equations with automatic domain decomposition, J. Comput. Phys., 144 (1998), 710-726.
doi: 10.1006/jcph.1998.6011. |
[32] |
S. Tiwari, A. Klar and S. Hardt, A particle-particle hybrid method for kinetic and continuum equations, J. Comput. Phys., 228 (2009), 7109-7124.
doi: 10.1016/j.jcp.2009.06.019. |
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