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

TL;DR

The paper tackles preserving moment realizability in hyperbolic quintuple-moment QBMM closures for kinetic equations, focusing on Gaussian-EQMOM and HyQMOM. It introduces a GQL-inspired reformulation that turns nonlinear realizability constraints into a nonnegative quadratic form in the moment vector and embeds these constraints into an HLL flux with closure-consistent wave speeds. The authors prove sufficient realizability-preserving CFL conditions for collisionless and BGK-relaxation regimes, augment the scheme with a limiter to enforce strict interfacial realizability, and demonstrate robust, oscillation-free performance across smooth and discontinuous problems, including low-density scenarios. The approach unifies realizability preservation for complex QBMM closures and accommodates multiscale transitions, with natural extension to higher-order discretizations and broader QBMM variants.

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.

Figures (8)

• Figure 1: Computed moments $\{ M_\ell(t=1,x)\}^{5}_{\ell=0}$ for Example \ref{['ex:1dRiemann']} with $\tau=\infty$ (no source term).
• Figure 2: Computed moments $\{M_\ell(t=1,x)\}^{5}_{\ell=0}$ for Example \ref{['ex:1dRiemann']} with $\tau=0.05$.
• Figure 3: Numerical results for Example \ref{['ex:1dShockTube']} at $t=0.2$ with $\tau=\infty$ (no source). (a): computed moments $\{M_\ell(t=1,x)\}_{\ell=0}^{5}$ computed with the two closures (each using its own initial data). (b): $\rho_1$, $\rho_2$, and $\sigma$ for the Gaussian-EQMOM system. (c): $\rho_1$, $\rho_2$, and $\rho_3$ for the HyQMOM system.
• Figure 4: Numerical results for Example \ref{['ex:1dShockTube']} at $t=0.2$ with $\tau=0.05$ . (a): computed moments $\{M_\ell(t=1,x)\}_{\ell=0}^{5}$ computed with the two closures (each using its own initial data). (b): $\rho_1$, $\rho_2$, and $\sigma$ for the Gaussian-EQMOM system. (c): $\rho_1$, $\rho_2$, and $\rho_3$ for the HyQMOM system.
• Figure 5: Numerical results of Example \ref{['ex:1dShuOsher']} at $t=1$ with $\tau=\infty$ (no source term). (a): computed moments $\{M_\ell(t=1,x)\}_{\ell=0}^{5}$ for the two closure systems (each with its own initial data). (b): $\rho_1$, $\rho_2$, and $\sigma$ for the Gaussian-EQMOM case. (c): $\rho_1$, $\rho_2$, and $\rho_3$ for the HyQMOM case.

Theorems & Definitions (22)

• Remark 2.1
• Theorem 2.2: Theorem 1.2 of shohat1943problem
• Lemma 3.1
• Proof 1
• Remark 3.2
• Lemma 3.3
• Proof 2
• Lemma 3.4
• Proof 3
• Theorem 3.5