American Institute of Mathematical Sciences

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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

Armando Majorana 1,

1. 

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

Received  September 2010 Revised  December 2010 Published  January 2011

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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Cercignani, "The Boltzmann Equation and its Applications,'', Springer, (1988).   Google Scholar

[6]

C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum, (1990).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

V. V. Aristov, "Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows,'', Kluwer Academic Publishers, (2001).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Cercignani, "The Boltzmann Equation and its Applications,'', Springer, (1988).   Google Scholar

[6]

C. Cercignani, "Mathematical Methods in Kinetic Theory,'', Plenum, (1990).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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