Provably realizability-preserving finite volume method for quadrature-based moment models of kinetic equations

TL;DR

This paper develops a provably realizability-preserving finite-volume numerical scheme for five-moment kinetic equations using quadrature-based closures, ensuring stability and accuracy across multiscale regimes.

Contribution

It introduces a novel flux construction and analysis that guarantees realizability preservation for complex hyperbolic moment systems with quadratic constraints.

Findings

The scheme maintains realizability under explicit CFL conditions.

Numerical tests confirm robustness and accuracy in low-density regions.

The method supports stiff-to-kinetic transitions and higher-order discretizations.

Abstract

Quadrature-based moment methods (QBMM) provide tractable closures for multiscale kinetic equations, with diverse applications across aerosols, sprays, and particulate flows, etc. However, for the derived hyperbolic moment-closure systems, seeking numerical schemes preserving moment realizability is essential yet challenging due to strong nonlinear coupling and the lack of explicit conservative-to-flux maps. This paper proposes and analyzes a provably realizability-preserving finite-volume method for five-moment systems closed by the two-node Gaussian-EQMOM and three-point HyQMOM. Rather than relying on kinetic fluxes, we recast the realizability condition into a nonnegative quadratic form in the moment vector, reducing the original nonlinear constraints to bilinear inequalities amenable to analysis. On this basis, we construct a tailored Harten--Lax--van Leer (HLL) flux with rigorously derived wave speeds and intermediate states that embed realizability directly into the flux evaluation. We prove sufficient realizability-preserving conditions under explicit Courant--Friedrichs--Lewy (CFL) constraints in the collisionless case, and for BGK relaxation, we obtain coupled time-step conditions involving a realizability radius; a semi-implicit BGK variant inherits the collisionless CFL. From a multiscale perspective, the analysis yields stability conditions uniform in the relaxation time and supports stiff-to-kinetic transitions. A practical limiter enforces strict realizability of reconstructed interface states without degrading accuracy. Numerical experiments demonstrate the accuracy, robustness in low-density regions, and realizability for both closures. This framework unifies realizability preservation for solving hyperbolic moment systems with complex closures and extends naturally to higher-order space--time discretizations.