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Stability analysis of quadrature-based moment methods for kinetic equations. (English) Zbl 1432.35159

Summary: In this paper, we give a systematic stability analysis of the quadrature-based moment method (QBMM) for the one-dimensional Boltzmann equation with BGK or Shakhov models. As reported in recent literature, the method has revealed its potential for modeling nonequilibrium flows, while a thorough theoretical analysis is largely missing but desirable. We show that the method can yield nonhyperbolic moment systems if the distribution function is approximated by a linear combination of \(\delta \)-functions. On the other hand, if the \(\delta \)-functions are replaced by their Gaussian approximations with a common variance, we prove that the moment systems are strictly hyperbolic and preserve the dissipation property (or \(H\)-theorem) of the kinetic equation. In the proof, we also determine the equilibrium manifold that lies on the boundary of the state space. The proofs are quite technical and involve detailed analyses of the characteristic polynomials of the coefficient matrices.

MSC:

35Q20 Boltzmann equations
35Q79 PDEs in connection with classical thermodynamics and heat transfer
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35B35 Stability in context of PDEs
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82M12 Finite volume methods applied to problems in statistical mechanics
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