Bounds-Preserving Lax-Wendroff Discontinuous Galerkin Schemes for Quadrature-Based Moment-Closure Approximations of Kinetic Models
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The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. A major challenge with these and other closures is that whenever the flux function must be evaluated (e.g., in a numerical update), a moment-inversion problem must be solved that computes the flux from the known input moments. In this work we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order Lax-Wendroff discontinuous Galerkin scheme for solving the resulting fluid system. The scheme is based on a predictor-corrector approach, where the prediction step is a localized space-time discontinuous Galerkin scheme. The nonlinear algebraic system that arises in this prediction step is solved using a Picard iteration. The correction step is a straightforward explicit update using the predicted solution in order to evaluate space-time flux integrals. In the absence of additional limiters, the proposed high-order scheme does not in general guarantee that the numerical solution remains in the convex set over which HyQMOM is moment-invertible. To overcome this challenge, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set over which moment-invertible and hyperbolicity of the fluid system is guaranteed. We develop positivity-preserving limiters in both the prediction and correction steps, as well as an oscillation-limiter that damps unphysical oscillations near shocks and rarefactions. Finally, we perform convergence tests to verify the order of accuracy of the scheme, as well as test the scheme on Riemann data to demonstrate the shock-capturing and robustness of the method.
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