Volume 17, Issue 2
On Singular Closures for the 5-Moment System in Kinetic Gas Theory

Roman Pascal Schaerer & Manuel Torrilhon

Commun. Comput. Phys., 17 (2015), pp. 371-400.

Published online: 2018-04

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  • Abstract

Moment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.
Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which not only suffer from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. A more recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.
Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

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@Article{CiCP-17-371, author = {}, title = {On Singular Closures for the 5-Moment System in Kinetic Gas Theory}, journal = {Communications in Computational Physics}, year = {2018}, volume = {17}, number = {2}, pages = {371--400}, abstract = {

Moment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.
Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which not only suffer from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. A more recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.
Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.201213.130814a}, url = {http://global-sci.org/intro/article_detail/cicp/10962.html} }
TY - JOUR T1 - On Singular Closures for the 5-Moment System in Kinetic Gas Theory JO - Communications in Computational Physics VL - 2 SP - 371 EP - 400 PY - 2018 DA - 2018/04 SN - 17 DO - http://doi.org/10.4208/cicp.201213.130814a UR - https://global-sci.org/intro/article_detail/cicp/10962.html KW - AB -

Moment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.
Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which not only suffer from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. A more recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.
Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

Roman Pascal Schaerer & Manuel Torrilhon. (2020). On Singular Closures for the 5-Moment System in Kinetic Gas Theory. Communications in Computational Physics. 17 (2). 371-400. doi:10.4208/cicp.201213.130814a
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