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March  2016, 9(1): 51-74. doi: 10.3934/krm.2016.9.51

## Asymptotic preserving scheme for a kinetic model describing incompressible fluids

 1 Inria Rennes Bretagne Atlantique (team IPSO) and IRMAR, University of Rennes 1, Campus de Beaulieu, 35042 Rennes, France 2 CNRS and IRMAR, University of Rennes 1, Campus de Beaulieu, 35042 Rennes, France 3 Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India 4 National Mathematics Initiative, Indian Institute of Science, Bangalore, India 5 Department of Civil Engineering, Indian Institute of Science, Bangalore, India

Received  September 2014 Revised  July 2015 Published  October 2015

The kinetic theory of fluid turbulence modeling developed by Degond and Lemou in [7] is considered for further study, analysis and simulation. Starting with the Boltzmann like equation representation for turbulence modeling, a relaxation type collision term is introduced for isotropic turbulence. In order to describe some important turbulence phenomenology, the relaxation time incorporates a dependency on the turbulent microscopic energy and this makes difficult the construction of efficient numerical methods. To investigate this problem, we focus here on a multi-dimensional prototype model and first propose an appropriate change of frame that makes the numerical study simpler. Then, a numerical strategy to tackle the stiff relaxation source term is introduced in the spirit of Asymptotic Preserving Schemes. Numerical tests are performed in a one-dimensional framework on the basis of the developed strategy to confirm its efficiency.
Citation: Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic and Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51
##### References:
 [1] M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032. [2] F. Bouchut and B. Perthame, The BGK model for small Prandtl number in the Navier-Stokes approximation, J. Stat. Phys., 71 (1993), 191-207. doi: 10.1007/BF01048094. [3] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529. [4] H. Chen, S. Kandasamy, S. Orzag, R. Shock, S. Succi and V. Yakhot, Extended Boltzmann kinetic equation for turbulent flows, Science Magazine, 301 (2003), 633-636. doi: 10.1126/science.1085048. [5] F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 26-42. doi: 10.1137/0728002. [6] 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, Kinetic Related Models, 4 (2011), 441-477. doi: 10.3934/krm.2011.4.441. [7] P. Degond and M. Lemou, Turbulence models for incompressible fluids derived from kinetic theory, J. Math. Fluid Mech., 4 (2002), 257-284. doi: 10.1007/s00021-002-8545-8. [8] P. Degond and M. Lemou, On viscosity and termal conduction of fluids with multivalued internal energy, Eur. J. Mech. B-Fluids, 20 (2001), 303-327. doi: 10.1016/S0997-7546(00)01095-5. [9] P. Degond and P. F. Peyrard, Un modèle de collisions ondes-particules en physique des plasmas: Application la dynamique des gaz, C. R. Acad. Sci. Paris, Ser I, 323 (1996), 209-214. [10] P. Degond, J. L. López and P. F. Peyrard, On the macroscopic dynamics induced by a model wave-particle collision operator, Continuum Mechanics and Thermodynamics, 10 (1998), 153-178. doi: 10.1007/s001610050087. [11] P. Degond, J.L. López, F. Poupaud and C. Schmeiser, Existence of solutions of a kinetic equation modeling cometary flows, J. Stat. Phys., 96 (1999), 361-376. doi: 10.1023/A:1004584719071. [12] G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for non-linear kinetic equations, SIAM Journal of Numerical Analysis, 51 (2013), 1064-1087. doi: 10.1137/12087606X. [13] G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM Journal of Numerical Analysis, 49 (2011), 2057-2077. doi: 10.1137/100811052. [14] J. Earl, J. R. Jokipii and G. Morfill, Cosmic ray viscosity, Astrophysical Journal, 331 (1988), L91-L94. doi: 10.1086/185242. [15] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. of Comput. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017. [16] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454. doi: 10.1137/S1064827598334599. [17] A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models, Math. Models Methods Appl. Sci., 11 (2001), 749-767. doi: 10.1142/S0218202501001082. [18] M. Lemou, Relaxed micro-macro schemes for kinetic equations, C.R. Acad. Sci. Paris, Ser. I, 348 (2010), 455-460. doi: 10.1016/j.crma.2010.02.017. [19] 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. Comput., 31 (2008), 334-368. doi: 10.1137/07069479X. [20] L. Mieussens and H. Struchtrup, Numerical comparison of BGK-models with proper Prandtl number, Phys. Fluids, 16 (2004), 2797-2813. [21] S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, Journal of Scientific Computing, 32 (2007), 1-28. doi: 10.1007/s10915-006-9116-6. [22] S. B. Pope, Turbulent Flows, Cambridge University Press, 2000. doi: 10.1017/CBO9780511840531. [23] H. Struchtrup, The BGK-model with velocity-dependent collision time, Cont. Mech. Thermodyn., 9 (1997), 23-31. doi: 10.1007/s001610050053. [24] D. C. Wilcox, Turbulence Modeling for CFD, D.C.W. Industries Inc., California, 1994.

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##### References:
 [1] M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032. [2] F. Bouchut and B. Perthame, The BGK model for small Prandtl number in the Navier-Stokes approximation, J. Stat. Phys., 71 (1993), 191-207. doi: 10.1007/BF01048094. [3] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529. [4] H. Chen, S. Kandasamy, S. Orzag, R. Shock, S. Succi and V. Yakhot, Extended Boltzmann kinetic equation for turbulent flows, Science Magazine, 301 (2003), 633-636. doi: 10.1126/science.1085048. [5] F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), 26-42. doi: 10.1137/0728002. [6] 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, Kinetic Related Models, 4 (2011), 441-477. doi: 10.3934/krm.2011.4.441. [7] P. Degond and M. Lemou, Turbulence models for incompressible fluids derived from kinetic theory, J. Math. Fluid Mech., 4 (2002), 257-284. doi: 10.1007/s00021-002-8545-8. [8] P. Degond and M. Lemou, On viscosity and termal conduction of fluids with multivalued internal energy, Eur. J. Mech. B-Fluids, 20 (2001), 303-327. doi: 10.1016/S0997-7546(00)01095-5. [9] P. Degond and P. F. Peyrard, Un modèle de collisions ondes-particules en physique des plasmas: Application la dynamique des gaz, C. R. Acad. Sci. Paris, Ser I, 323 (1996), 209-214. [10] P. Degond, J. L. López and P. F. Peyrard, On the macroscopic dynamics induced by a model wave-particle collision operator, Continuum Mechanics and Thermodynamics, 10 (1998), 153-178. doi: 10.1007/s001610050087. [11] P. Degond, J.L. López, F. Poupaud and C. Schmeiser, Existence of solutions of a kinetic equation modeling cometary flows, J. Stat. Phys., 96 (1999), 361-376. doi: 10.1023/A:1004584719071. [12] G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for non-linear kinetic equations, SIAM Journal of Numerical Analysis, 51 (2013), 1064-1087. doi: 10.1137/12087606X. [13] G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations, SIAM Journal of Numerical Analysis, 49 (2011), 2057-2077. doi: 10.1137/100811052. [14] J. Earl, J. R. Jokipii and G. Morfill, Cosmic ray viscosity, Astrophysical Journal, 331 (1988), L91-L94. doi: 10.1086/185242. [15] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. of Comput. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017. [16] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454. doi: 10.1137/S1064827598334599. [17] A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models, Math. Models Methods Appl. Sci., 11 (2001), 749-767. doi: 10.1142/S0218202501001082. [18] M. Lemou, Relaxed micro-macro schemes for kinetic equations, C.R. Acad. Sci. Paris, Ser. I, 348 (2010), 455-460. doi: 10.1016/j.crma.2010.02.017. [19] 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. Comput., 31 (2008), 334-368. doi: 10.1137/07069479X. [20] L. Mieussens and H. Struchtrup, Numerical comparison of BGK-models with proper Prandtl number, Phys. Fluids, 16 (2004), 2797-2813. [21] S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations, Journal of Scientific Computing, 32 (2007), 1-28. doi: 10.1007/s10915-006-9116-6. [22] S. B. Pope, Turbulent Flows, Cambridge University Press, 2000. doi: 10.1017/CBO9780511840531. [23] H. Struchtrup, The BGK-model with velocity-dependent collision time, Cont. Mech. Thermodyn., 9 (1997), 23-31. doi: 10.1007/s001610050053. [24] D. C. Wilcox, Turbulence Modeling for CFD, D.C.W. Industries Inc., California, 1994.
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