September  2017, 10(3): 643-668. doi: 10.3934/krm.2017026

Asymptotic preserving and time diminishing schemes for rarefied gas dynamic

1. 

INRIA-Rennes Bretagne Atlantique, IPSO Project, and IRMAR (Université de Rennes 1), 35042 Rennes, France

2. 

Department of Mathematics and Computer Science, University of Ferrara, 44121 Ferrara, Italy

3. 

CNRS and IRMAR (Université de Rennes 1 and ENS Rennes), and INRIA-Rennes Bretagne Atlantique, IPSO Project, 35042 Rennes, France

Received  June 2016 Revised  September 2016 Published  December 2016

Fund Project: The authors have been supported by the ANR project Moonrise

In this work, we introduce a new class of numerical schemes for rarefied gas dynamic problems described by collisional kinetic equations. The idea consists in reformulating the problem using a micro-macro decomposition and successively in solving the microscopic part by using asymptotic preserving Monte Carlo methods. We consider two types of decompositions, the first leading to the Euler system of gas dynamics while the second to the Navier-Stokes equations for the macroscopic part. In addition, the particle method which solves the microscopic part is designed in such a way that the global scheme becomes computationally less expensive as the solution approaches the equilibrium state as opposite to standard methods for kinetic equations which computational cost increases with the number of interactions. At the same time, the statistical error due to the particle part of the solution decreases as the system approach the equilibrium state. This causes the method to degenerate to the sole solution of the macroscopic hydrodynamic equations (Euler or Navier-Stokes) in the limit of infinite number of collisions. In a last part, we will show the behaviors of this new approach in comparisons to standard Monte Carlo techniques for solving the kinetic equation by testing it on different problems which typically arise in rarefied gas dynamic simulations.

Citation: 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
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. Google Scholar

[2]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ. Formal derivation, J. Statist. Phys., 63 (1991), 323-344. doi: 10.1007/BF01026608. Google Scholar

[3]

M. BennouneM. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032. Google Scholar

[4] G. A. Bird, Molecular Gas Dynamics and Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1995.
[5]

S. Brunner, E. Valeo and J. A. Krommes, Collisional Delta-F Scheme with Evolving Background for Transport Time Scale Simulations, Phys. of Plasmas, 1999.Google Scholar

[6]

S. Brunner, E. Valeo and J. A. Krommes, Linear Delta-F Simulations of Nonlocal Electron Heat Transport, Phys. of Plasmas, 2000.Google Scholar

[7]

J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation, J. Comput. Phys., 228 (2009), 460-475. Google Scholar

[8]

R. CaflischC. WangG. DimarcoB. Cohen and A. Dimits, A hybrid method for accelerated simulation of Coulomb collisions in a plasma, SIAM MMS, 7 (2008), 865-887. doi: 10.1137/070704939. Google Scholar

[9]

R. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804. Google Scholar

[10]

M. Campos Pinto and F. Charles, Uniform convergence of a linearly transformed particle method for the Vlasov-Poisson system, SIAM J. Numer. Anal., 54 (2016), 137-160. doi: 10.1137/140994678. Google Scholar

[11] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.
[12]

A. CrestettoN. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, Kin. Rel. Models, 5 (2012), 787-816. doi: 10.3934/krm.2012.5.787. Google Scholar

[13]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: Diffusion and high-field scaling limits, Kin. Rel. Models, 4 (2011), 441-477. doi: 10.3934/krm.2011.4.441. Google Scholar

[14]

P. DegondG. Dimarco and L. Pareschi, The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids, 67 (2011), 189-213. doi: 10.1002/fld.2345. Google Scholar

[15]

P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct simulation Monte-Carlo, J. Comput. Phys., 231 (2012), 2414-2437. doi: 10.1016/j.jcp.2011.11.030. Google Scholar

[16]

P. DegondS. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys., 209 (2005), 665-694. doi: 10.1016/j.jcp.2005.03.025. Google Scholar

[17]

G. Dimarco, The hybrid moment guided Monte Carlo method for the Boltzmann equation, Kin. Rel. Models, 6 (2013), 291-315. doi: 10.3934/krm.2013.6.291. Google Scholar

[18]

G. Dimarco and L. Pareschi, Hybrid multiscale methods Ⅱ. Kinetic equations, SIAM MMS, 6 (2007), 1169-1197. doi: 10.1137/070680916. Google Scholar

[19]

G. Dimarco and L. Pareschi, A fluid solver independent hybrid method for multiscale kinetic equations, SIAM J. Sci. Comput., 32 (2010), 603-634. doi: 10.1137/080730585. Google Scholar

[20]

G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for non linear kinetic equations, SIAM J. Num. Anal., 51 (2013), 1064-1087. doi: 10.1137/12087606X. Google Scholar

[21]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520. doi: 10.1017/S0962492914000063. Google Scholar

[22]

F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multiscale kinetic equations, J. Sci. Comp., 37 (2015), A1218-A1247. doi: 10.1137/140958773. Google Scholar

[23]

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. Google Scholar

[24]

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. Google Scholar

[25]

T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701. Google Scholar

[26]

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

[27]

Q. LiJ. Lu and W. Sun, Diffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics, J. Comp. Phys., 292 (2015), 141-167. doi: 10.1016/j.jcp.2015.03.014. Google Scholar

[28]

Q. Li, J. Lu and W. Su, Half-space Kinetic Equations with General Boundary Conditions, to appear in Math. Comp., 2016.Google Scholar

[29]

M. Lemou, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Mathématique, 348 (2010), 455-460. doi: 10.1016/j.crma.2010.02.017. Google Scholar

[30]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comp., 31 (2008), 334-368. doi: 10.1137/07069479X. Google Scholar

[31]

P. LeTallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes, J. Comput. Phys., 136 (1997), 51-67. doi: 10.1006/jcph.1997.5729. Google Scholar

[32]

R. J. Leveque, Numerical Methods for Conservation Laws, Lecture in Mathematics, Birkhåuser, 1992. doi: 10.1007/978-3-0348-8629-1. Google Scholar

[33]

Q. LiJ. Lu and W. Sun, Diffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics, J. Comp. Phys., 292 (2015), 141-167. doi: 10.1016/j.jcp.2015.03.014. Google Scholar

[34]

Q. Li, J. Lu and W. Su, Half-space kinetic equations with general boundary conditions, to appear in Math. Comp. (2016).Google Scholar

[35]

S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2004.Google Scholar

[36]

K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), 2042-2049. Google Scholar

[37]

G. A. RadtkeJ.-P. M. Peraud and N. Hadjiconstantinou, On efficient simulations of multiscale kinetic transport, Phil. Trans. Royal Soc. A, 371 (2013), 20120182, 19pp. doi: 10.1098/rsta.2012.0182. Google Scholar

[38]

S. TiwariA. 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. Google Scholar

[39]

B. Yan, A hybrid method with deviational particles for spatial inhomogeneous plasma, J. Comput. Phys., 309 (2016), 18-36. doi: 10.1016/j.jcp.2015.12.050. Google Scholar

[40]

B. Yan and R. Caflisch, A Monte Carlo method with negative particles for Coulomb collisions, J. Comput. Phys., 298 (2015), 711-740. doi: 10.1016/j.jcp.2015.06.021. Google Scholar

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. Google Scholar

[2]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ. Formal derivation, J. Statist. Phys., 63 (1991), 323-344. doi: 10.1007/BF01026608. Google Scholar

[3]

M. BennouneM. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032. Google Scholar

[4] G. A. Bird, Molecular Gas Dynamics and Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1995.
[5]

S. Brunner, E. Valeo and J. A. Krommes, Collisional Delta-F Scheme with Evolving Background for Transport Time Scale Simulations, Phys. of Plasmas, 1999.Google Scholar

[6]

S. Brunner, E. Valeo and J. A. Krommes, Linear Delta-F Simulations of Nonlocal Electron Heat Transport, Phys. of Plasmas, 2000.Google Scholar

[7]

J. Burt and I. Boyd, A hybrid particle approach for continuum and rarefied flow simulation, J. Comput. Phys., 228 (2009), 460-475. Google Scholar

[8]

R. CaflischC. WangG. DimarcoB. Cohen and A. Dimits, A hybrid method for accelerated simulation of Coulomb collisions in a plasma, SIAM MMS, 7 (2008), 865-887. doi: 10.1137/070704939. Google Scholar

[9]

R. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804. Google Scholar

[10]

M. Campos Pinto and F. Charles, Uniform convergence of a linearly transformed particle method for the Vlasov-Poisson system, SIAM J. Numer. Anal., 54 (2016), 137-160. doi: 10.1137/140994678. Google Scholar

[11] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.
[12]

A. CrestettoN. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, Kin. Rel. Models, 5 (2012), 787-816. doi: 10.3934/krm.2012.5.787. Google Scholar

[13]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: Diffusion and high-field scaling limits, Kin. Rel. Models, 4 (2011), 441-477. doi: 10.3934/krm.2011.4.441. Google Scholar

[14]

P. DegondG. Dimarco and L. Pareschi, The moment guided Monte Carlo method, International Journal for Numerical Methods in Fluids, 67 (2011), 189-213. doi: 10.1002/fld.2345. Google Scholar

[15]

P. Degond and G. Dimarco, Fluid simulations with localized Boltzmann upscaling by direct simulation Monte-Carlo, J. Comput. Phys., 231 (2012), 2414-2437. doi: 10.1016/j.jcp.2011.11.030. Google Scholar

[16]

P. DegondS. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys., 209 (2005), 665-694. doi: 10.1016/j.jcp.2005.03.025. Google Scholar

[17]

G. Dimarco, The hybrid moment guided Monte Carlo method for the Boltzmann equation, Kin. Rel. Models, 6 (2013), 291-315. doi: 10.3934/krm.2013.6.291. Google Scholar

[18]

G. Dimarco and L. Pareschi, Hybrid multiscale methods Ⅱ. Kinetic equations, SIAM MMS, 6 (2007), 1169-1197. doi: 10.1137/070680916. Google Scholar

[19]

G. Dimarco and L. Pareschi, A fluid solver independent hybrid method for multiscale kinetic equations, SIAM J. Sci. Comput., 32 (2010), 603-634. doi: 10.1137/080730585. Google Scholar

[20]

G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for non linear kinetic equations, SIAM J. Num. Anal., 51 (2013), 1064-1087. doi: 10.1137/12087606X. Google Scholar

[21]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520. doi: 10.1017/S0962492914000063. Google Scholar

[22]

F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multiscale kinetic equations, J. Sci. Comp., 37 (2015), A1218-A1247. doi: 10.1137/140958773. Google Scholar

[23]

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. Google Scholar

[24]

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. Google Scholar

[25]

T. Homolle and N. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701. Google Scholar

[26]

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

[27]

Q. LiJ. Lu and W. Sun, Diffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics, J. Comp. Phys., 292 (2015), 141-167. doi: 10.1016/j.jcp.2015.03.014. Google Scholar

[28]

Q. Li, J. Lu and W. Su, Half-space Kinetic Equations with General Boundary Conditions, to appear in Math. Comp., 2016.Google Scholar

[29]

M. Lemou, Relaxed micro-macro schemes for kinetic equations, Comptes Rendus Mathématique, 348 (2010), 455-460. doi: 10.1016/j.crma.2010.02.017. Google Scholar

[30]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comp., 31 (2008), 334-368. doi: 10.1137/07069479X. Google Scholar

[31]

P. LeTallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes by half fluxes, J. Comput. Phys., 136 (1997), 51-67. doi: 10.1006/jcph.1997.5729. Google Scholar

[32]

R. J. Leveque, Numerical Methods for Conservation Laws, Lecture in Mathematics, Birkhåuser, 1992. doi: 10.1007/978-3-0348-8629-1. Google Scholar

[33]

Q. LiJ. Lu and W. Sun, Diffusion approximations and domain decomposition method of linear transport equations: Asymptotics and numerics, J. Comp. Phys., 292 (2015), 141-167. doi: 10.1016/j.jcp.2015.03.014. Google Scholar

[34]

Q. Li, J. Lu and W. Su, Half-space kinetic equations with general boundary conditions, to appear in Math. Comp. (2016).Google Scholar

[35]

S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2004.Google Scholar

[36]

K. Nanbu, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), 2042-2049. Google Scholar

[37]

G. A. RadtkeJ.-P. M. Peraud and N. Hadjiconstantinou, On efficient simulations of multiscale kinetic transport, Phil. Trans. Royal Soc. A, 371 (2013), 20120182, 19pp. doi: 10.1098/rsta.2012.0182. Google Scholar

[38]

S. TiwariA. 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. Google Scholar

[39]

B. Yan, A hybrid method with deviational particles for spatial inhomogeneous plasma, J. Comput. Phys., 309 (2016), 18-36. doi: 10.1016/j.jcp.2015.12.050. Google Scholar

[40]

B. Yan and R. Caflisch, A Monte Carlo method with negative particles for Coulomb collisions, J. Comput. Phys., 298 (2015), 711-740. doi: 10.1016/j.jcp.2015.06.021. Google Scholar

Figure 1.  Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-2}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Unsteady Shock test
Figure 2.  Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-3}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Unsteady Shock test
Figure 3.  Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-4}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Unsteady Shock test
Figure 4.  Left: time evolution of the number of effective particles (in semi-logarithmic scale) used in the Asymptotic Preserving Time Diminishing (Euler) method and in the Monte Carlo method for different values of the Knudsen number ($\varepsilon=10^{-2}, 5\cdot 10^{-3}, 10^{-3}, 5\cdot 10^{-4}, 10^{-4}$). Middle: time evolution of the ratio of the number of particles used for APTD versus the number of particles for the corresponding MC simulation for different values of $\varepsilon $. Right: time evolution of the ratio of the number of particles used for APTD versus the number of particles for APTDNS for different values of $\varepsilon $. Unsteady Shock test
Figure 5.  Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-2}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Sod test
Figure 6.  Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-3}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Sod test
Figure 7.  Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-4}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Sod test
Figure 8.  Left: time evolution of the number of effective particles (in semi-logarithmic scale) used in the Asymptotic Preserving Time Diminishing (Euler) method and in the Monte Carlo method for different values of the Knudsen number ($\varepsilon=10^{-2}, 5\cdot 10^{-3}, 10^{-3}, 5\cdot 10^{-4}, 10^{-4}$). Middle: time evolution of the ratio of the number of particles used for APTD versus the number of particles for the corresponding MC simulation for different values of $\varepsilon $. Right: time evolution of the ratio of the number of particles used for APTD versus the number of particles for APTDNS for different values of $\varepsilon $. Sod test
Figure 9.  Error ($L^1$ norm) for the density, the mean velocity and the temperature for the Asymptotic Preserving Time Diminishing NS method (left column), and for the Monte Carlo method (right column) for different values of $\varepsilon $ (from top to bottom, $\varepsilon =10^{-2}, 10^{-3}, 10^{-4}$)
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