A locally-implicit Lax-Wendroff discontinuous Galerkin scheme with limiters that guarantees moment-realizability for quadrature-based moment closures
A polydisperse multiphase flow is one in which one of the thermodynamic phases can be considered discrete while the other is assumed to be continuous. Examples include sprays and bubbly flows. Such flows can be modeled by kinetic models that describe the evolution of a probability density function (PDF) in phase space. Moments of this PDF describe physical observable quantities such as the mass, momentum, and energy in
the flow. From a computational perspective, kinetic models are expensive to solve since they require the resolution of a very high-dimensional phase space. In many polydisperse multiphase flow applications, the phase space would consist of the spatial variables (three dimensions), the velocity variables (three dimensions), a size dimension (one dimension), and time (one dimension).
An alternative to kinetic models are so-called fluid models, which instead of evolving the full kinetic distribution function, evolve only a small subset of moments of the PDF. Fluid models are defined on a lower dimensional space than the original distribution function, and as such, are computationally much more tractable. The difficulty with fluid models is that the precise form of the fluid approximation depends on the choice of the moment-closure. In general, finding a suitable robust moment-closure is still an open scientific problem.
In this work, we consider an approach for developing fluid approximations to kinetic equations known as quadrature-based moment-closures. The true distribution function is replaced by a finite set of Dirac delta functions with variable weights and abscissas. After developing this model, we then propose a high-order numerical method using the discontinuous Galerkin (DG) finite element method to discretize the resulting system.
In particular, we develop a Lax-Wendroff discontinuous Galerkin method, which allows us to efficiently achieve high-order through a locally-implicit prediction step, followed by a fully explicit correction step. The key difficulty in applying high-order schemes to nonlinear hyperbolic systems is that the formation of shocks, rarefactions, and vacuum states cause the numerical solution to produce large unphysical oscillations that lead to
simulation failures. To remedy these problems, we develop a series of post-processing steps, referred to as limiters, that provably guarantee that the solution remains physical and that suppress unphysical solutions. The resulting method is verified on several shock tube problems.