A numerical model of the Boltzmann equation related to the discontinuous Galerkin method

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March 2011, 4(1): 139-151. doi: 10.3934/krm.2011.4.139

A numerical model of the Boltzmann equation related to the discontinuous Galerkin method

1.	Dipartimento di Matematica e Informatica, Viale A. Doria 6, 95125 Catania, Italy

Received September 2010 Revised December 2010 Published January 2011

Abstract
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We propose a new deterministic numerical model, based on the discontinuous Galerkin method, for solving the nonlinear Boltzmann equation for rarefied gases. A set of partial differential equations is derived and analyzed. The new model guarantees the conservation of the mass, momentum and energy for homogeneous solutions. We avoid any stochastic procedures in the treatment of the collision operator of the Boltzmamn equation.

Keywords: discontinuous Galerkin finite elements., Boltzmann equation.

Mathematics Subject Classification: Primary: 76P, 82C40; Secondary: 65M6.

Citation: Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139

References:

[1]	V. V. Aristov, "Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows,'', Kluwer Academic Publishers, (2001).
[2]	L. L. Baker and N. G. Hadjiconstantinou, Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: A discontinuous Galerkin formulation,, Int. J. Numer. Meth. Fluids, 58 (2008), 381. doi: 10.1002/fld.1724.
[3]	J. A. Carrillo, I. M. Gamba, A. Majorana and C.-W. Shu, A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices. Performance and comparisons with Monte Carlo methods,, Journal of Computational Physics, 184 (2003), 498. doi: 10.1016/S0021-9991(02)00032-3.
[4]	J. A. Carrillo, I. M. Gamba, A. Majorana and C.-W. Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: Efficiency, boundary conditions and comparison to Monte Carlo methods,, Journal of Computational Physics, 214 (2006), 55. doi: 10.1016/j.jcp.2005.09.005.
[5]	C. Cercignani, "The Boltzmann Equation and its Applications,'', Springer, (1988).
[6]	C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum, (1990).
[7]	Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann-Poisson systems for semiconductor devices,, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 3130. doi: 10.1016/j.cma.2009.05.015.
[8]	B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems,, Journal of Scientific Computing, 16 (2001), 173. doi: 10.1023/A:1012873910884.
[9]	F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation,, Transport Theory Statist. Phys., 23 (1994), 313. doi: 10.1080/00411459408203868.
[10]	Y. Sone, T. Ohwada and K. Aoki, Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules,, Physics of Fluids A, 1 (1989), 363. doi: 10.1063/1.857457.

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References:

[1]	V. V. Aristov, "Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows,'', Kluwer Academic Publishers, (2001).
[2]	L. L. Baker and N. G. Hadjiconstantinou, Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: A discontinuous Galerkin formulation,, Int. J. Numer. Meth. Fluids, 58 (2008), 381. doi: 10.1002/fld.1724.
[3]	J. A. Carrillo, I. M. Gamba, A. Majorana and C.-W. Shu, A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices. Performance and comparisons with Monte Carlo methods,, Journal of Computational Physics, 184 (2003), 498. doi: 10.1016/S0021-9991(02)00032-3.
[4]	J. A. Carrillo, I. M. Gamba, A. Majorana and C.-W. Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: Efficiency, boundary conditions and comparison to Monte Carlo methods,, Journal of Computational Physics, 214 (2006), 55. doi: 10.1016/j.jcp.2005.09.005.
[5]	C. Cercignani, "The Boltzmann Equation and its Applications,'', Springer, (1988).
[6]	C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum, (1990).
[7]	Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann-Poisson systems for semiconductor devices,, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 3130. doi: 10.1016/j.cma.2009.05.015.
[8]	B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems,, Journal of Scientific Computing, 16 (2001), 173. doi: 10.1023/A:1012873910884.
[9]	F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation,, Transport Theory Statist. Phys., 23 (1994), 313. doi: 10.1080/00411459408203868.
[10]	Y. Sone, T. Ohwada and K. Aoki, Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules,, Physics of Fluids A, 1 (1989), 363. doi: 10.1063/1.857457.

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