A parallel, implicit, adaptive mesh refinement (AMR), finite-volume scheme is described for the solution of the regularized Gaussian moment closure. The latter incorporates the influences of heat transfer by means of a first-order correction to the standard Gaussian closure. The combined moment closure treatment / numerical method is applied to the prediction of three-dimensional, non-equilibrium, micro-scale, gaseous flows. Unlike other regularized moment closures, the underlying maximum-entropy Gaussian closure provides a fully-realizable and strictly hyperbolic description of non-equilibrium gaseous flows that is valid from the continuum limit, through the transition regime, and up to the near-collisionless, free-molecular flow limit. The regularized closure provides a similarly robust description than now includes a fully anisotropic description of heat transfer. The proposed finite-volume scheme makes use of Riemann-solver-based flux functions and limited linear reconstruction to provide accurate and monotonic solutions, even in the presence of large solution gradients and/or under-resolved solution content on three-dimensional, multi-block, body-fitted, hexahedral mesh. A rather effective and highly scalable parallel implicit time-marching scheme based on a Jacobian-free inexact Newton-Krylov-Schwarz (NKS) approach with additive Schwarz preconditioning and domain partitioning following from the multi-block AMR mesh is used to obtain solutions to the non-linear ordinary-differential equations that result from finite-volume spatial discretization procedure. Details are given of the regularized Gaussian closure, with suitable extensions for diatomic gases, and slip-flow boundary treatment. Numerical results for several canonical flow problems demonstrate the potential of the regularized closures, that when combined with an efficient parallel solution method, provide and effective means for accurately predicting a range of fully three-dimensional non-equilibrium gaseous flow behavior.
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