Higher order hyperbolic quadrature method of moments for solving kinetic equations

This work attempts to solve two fundamental problems that can be encountered when solving kinetic equations using quadrature based moment methods (QBMM). These two problems are considered crucial problems since they are observed even in the simplest particulate systems. On the one hand, we have that particle trajectory crossing (PTC) events can lead to non-physical accumulation of particles in the solution when the system of transport equations is weakly hyperbolic (i.e. delta-shock formation). On the other hand, the fact that particle segregation events can create moment sets close to the frontier of moment space. These, in turn, can produce non-physical high-velocity fluxes that will significantly reduce the computational efficiency. Given that both problems are fundamental, we limit the framework to one-dimensional free transport of particles. For the first problem we create a novel hyperbolic quadrature moment closure. Although we lack formal analytic proof for a quadrature order (N) higher than three, the numerical examples presented clearly show a well-behaved solution, absent of delta-shocks up to an eight-order quadrature. Our solution to the second problem consists of two minor, though effective, modifications to the general numerical algorithm frequently used for solving kinetic equations using QBMM. The performance of the new algorithm is remarkably stable and efficient, shown by a numerical example.

Publication year:2021

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