Development of high-order realizable finite-volume schemes for quadrature-based moment method

Kinetic equations containing terms for spatial transport, gravity, fluid drag and particleparticle collisions can be used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the problem in terms of moments of velocity distribution. Recently, a quadrature-based moment method was derived by Fox for approximating solutions to kinetic equation for arbitrary Knudsen number. Fox also described 1and 2-order finite-volume schemes for solving the equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from moments. The moment-inversion algorithm does not work if the moments are non-realizable, meaning they do not correspond to a distribution function. Not all the finite-volume schemes lead to realizable moments. Desjardins et al. showed that realizability is guaranteed with the 1-order finite-volume scheme, but at the expense of excess numerical diffusion. In the present work, the nonrealizability of the standard 2-order finite-volume scheme is demonstrated and a generalized idea for the development of high-order realizable finite-volume schemes for quadrature-based moment methods is presented. This marks a significant improvement in the accuracy of solutions using the quadrature-based moment method as the use of 1-order scheme to guarantee realizability is no longer a limitation.