| A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion |
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| 作者姓名： | Shi Jin Lorenzo Pareschi Marshall Slemrod |
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| 作者单位： | Department of Mathematics，University of Wisconsin，Madison，Van Vleck Hall，WI 53706，USA，Department of Mathematics，University of Ferrara，Via Machiavelli 35，I-44100，Italy & Department of Mathematics，University of Wisconsin-Madison，Van Vleck Hall，WI 53706，USA.，Department of Mathematics，University of Wisconsin-Madison，Madison，WI 53715-1149，USA. |
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| 基金项目： | Supported by NSF grant DMS-0196106，Supported by NSF grant DMS-9803223 and DMS-00711463. |
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| 摘 要： | Abstract In 16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jinand Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weaklyparabolic, has a linearly hyperbolic convection part, and is endowed with a generalized eotropy inequality. Itagrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system forthe Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme,and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and bythe extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments showthat the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equationobtained by the DSMC, for a range of Mach numbers for hypersonic flows, th
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A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion |
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| Shi Jin,Lorenzo Pareschi,Marshall Slemrod.A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion[J].Acta Mathematicae Applicatae Sinica,2002,18(1):37-62. |
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| Authors: | Shi Jin Lorenzo Pareschi Marshall Slemrod |
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| Institution: | (1) Department of Mathematics, University of Visconsin, Madison, Van Vleck Hall, WI 53706, USA. (E-mail: Jin@math.wisc.edu), US;(2) Department of Mathematics, University of Ferrara, Via Machiavelli 35, I-44100, Italy & Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, WI 53706, USA. (E-mail: pareschi@dm.unife.it),;(3) Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53715-1149, USA. (E-mail: slemrod@math.wisc.edu), US |
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| Abstract: | In 16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems. |
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| Keywords: | Boltzmann equation Chapman-Enskog expansion Burnett equations relaxation central schemes |
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