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Volume 15, Issue 4
Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

Qin Li & Xu Yang

DOI: 10.4208/cicp.010113.160813s

Commun. Comput. Phys., 15 (2014), pp. 996-1011.

Published online: 2014-04

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  • Abstract

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 to, 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.

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COPYRIGHT: © Global Science Press

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@Article{CiCP-15-996, author = {Qin Li and Xu Yang}, title = {Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation}, journal = {Communications in Computational Physics}, year = {2014}, volume = {15}, number = {4}, pages = {996--1011}, abstract = {

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 to, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.010113.160813s}, url = {http://global-sci.org/intro/article_detail/cicp/7124.html} }
TY - JOUR T1 - Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation AU - Qin Li & Xu Yang JO - Communications in Computational Physics VL - 4 SP - 996 EP - 1011 PY - 2014 DA - 2014/04 SN - 15 DO - http://doi.org/10.4208/cicp.010113.160813s UR - https://global-sci.org/intro/article_detail/cicp/7124.html KW - AB -

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 to, 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.

Qin Li and Xu Yang. (2014). Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation. Communications in Computational Physics. 15 (4). 996-1011. doi:10.4208/cicp.010113.160813s
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