Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation
Qin Li 1, Xu Yang 2*1 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.
2 Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA.
Received 1 January 2013; Accepted (in revised version) 16 August 2013
Available online 21 January 2014
This paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi, SIAM Numer. Anal., 49 (2011), pp. 2057-2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied on, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.AMS subject classifications: 65C20, 65M06, 76D05, 82C40
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Key words: Multispecies Boltzmann equation, exponential Runge-Kutta method, hydrodynamic limit, asymptotic preserving property, positivity preserving.
Email: email@example.com (Q. Li), firstname.lastname@example.org (X. Yang)