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STABLE BOUNDARY CONDITIONS AND DISCRETIZATION FOR P_(N) EQUATIONS

Journal of Computational Mathematics 2022年第6期40卷 977-1003页

作者： Jonas Bunger Neeraj Sarna manuel torrilhon Applied and Computational Mathematics(ACoM) Department of MathematicsRWTH Aachen UniversitySchinkelstr 252062 AachenGermany

A solution to the linear Boltzmann equation satisfies an energy bound,which reflects a natural fact:The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time.In this paper,we present boundary conditions(BCs)for the spherical harmonic(P_(N))approximation,which ensure that this fundamental energy bound is satisfied by the P_(N) approximation.Our BCs are compatible with the characteristic waves of P_(N) equations and determine the incoming waves uniquely.Both,energy bound and compatibility,are shown on abstract formulations of P_(N) equations and BCs to isolate the necessary structures and properties.The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step,which is similar to the truncation of the series expansion in the P_(N) method.We show that summation by parts(SBP)finite differences on staggered grids in space and the method of simultaneous approximation terms(SAT)allows to maintain the energy bound also on the semi-discrete level.

关键词： Boundary conditions Energy stability Spherical harmonic(P_(N))approximation Kinetic theory Moment method Boltzmann Linear transport

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Hyperbolic Moment Equations in Kinetic Gas Theory Based on Multi-Variate Pearson-IV-Distributions

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Communications in Computational Physics 2010年第4期7卷 639-673页

作者： manuel torrilhon Seminar for Applied Mathematics ETH ZurichSwitzerland

In this paper we develop a new closure theory for moment approximationsin kinetic gas theory and derive hyperbolic moment equations for 13 fluid variablesincluding stress and heat flux. Classical equations have either restricted hyperbolicity regions like Grad’s moment equations or fail to include higher moments in apractical way like the entropy maximization approach. The new closure is based onPearson-Type-IV distributions which reduce to Maxwellians in equilibrium, but allowanisotropies and skewness in non-equilibrium. The closure relations are essentiallyexplicit and easy to evaluate. Hyperbolicity is shown numerically for a large range ofvalues. Numerical solutions of Riemann problems demonstrate the capability of thenew equations to handle strong non-equilibrium.

关键词： Kinetic gas theory moment approximations hyperbolic partial differential equations.

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