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Commun. Comput. Phys., 17 (2015), pp. 371-400.
Published online: 2018-04
Cited by
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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.
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.