Multiscale Computations for Kinetic and Related Problems

Hyperbolic Quadrature Method of Moments

Rodney Fox

Iowa State University


The quadrature method of moments (QMOM) reconstructs a velocity distribution function (VDF) from its integer moments: ${ M_0, M_1, dots, M_{2N-1} }$. The reconstructed VDF is a sum of weighted Dirac delta functions in phase space, and closes the spatial flux ($M_{2N}$) in the kinetic equation. The QMOM closure for $M_{2N}$ leads to a emph{weakly hyperbolic} system of moment equations. Here, we present an alternative closure where the moment $M_{5}$ is a function of ${ M_0, M_1, dots, M_{4} }$ chosen such that the five-moment system is hyperbolic. We refer to the VDF reconstruction with this choice for $M_{5}$ as the emph{hyperbolic quadrature method of moments} (HyQMOM) reconstruction. For HyQMOM, we show that (1) a choice for $M_{5}$ exists that is valid for realizable moments ${ M_0, M_1, M_2, M_3, M_4 }$, (2) the five eigenvalues of the moment system can be computed explicitly, and (3) the kinetic-based (KB) flux for the system depends on four of the five eigenvalues. In the limit where $M_4$ is on the boundary of moment space, the KB flux reduces to the 2-node QMOM flux, while for Gaussian moments it corresponds to a 4-node Gauss--Hermite quadrature. A 1-D Riemann problem is solved with HyQMOM to illustrate its ability to handle non-equilibrium VDF without creating delta shocks. For a multi-variate VDF, a hyperbolic modification of the conditional quadrature method of moments (CHyQMOM) has been developed. For example, in 2-D phase space bivariate moments (i.e. $M_{i , j} : 0 le i + j le 3 , (i , j) in (4,0) , (0,4)$) can be controlled thanks to a judicious choice of the nine velocity abscissas. CHyQMOM reconstructions for moments $M_{i , j, k}$ employ 27 velocity abscissas. The KB fluxes in 2/3-D are defined using the 1-D eigenvalues and directional splitting. Results for 2-D and 3-D crossing jets flows solved with CHyQMOM are presented to demonstrate its ability to capture binary crossing without dispersion.