Constrained-Optimization Based Data Transfer

We formulate a new class of optimization-based methods for data transfer (remap) of a scalar conserved quantity between two close meshes with the same connectivity. We present the methods in the context of the remap of a mass density field, which preserves global mass (the integral of the density over the computational domain). The key idea is to formulate remap as a global inequality-constrained optimization problem for mass fluxes between neighboring cells. The objective is to minimize the discrepancy between these fluxes and the given high-order target mass fluxes, subject to constraints that enforce physically motivated bounds on the associated primitive variable. In so doing, we separate accuracy considerations, handled by the objective functional, from the enforcement of physical bounds, handled by the constraints. The resulting second-order, conservative, and bound-preserving optimization-based remap (OBR) formulation is applicable to general, unstructured, heterogeneous grids. Under some weak requirements on grid proximity we prove that the OBR algorithm preserves linear fields in one, two and three dimensions. The chapter also examines connections between the OBR and the flux-corrected remap (FCR), which can be interpreted as a modified version of OBR (M-OBR), with the same objective but a smaller feasible set. The feasible set for M-OBR (FCR) is given by simple box constraints derived by using a “worst-case” scenario approach, which may result in loss of linearity preservation and ultimately accuracy for some grid motions. The optimality of the OBR solution means that, given a set of target fluxes and a distance measure, OBR finds the best possible approximations of these fluxes with respect to this measure, which also satisfy the physically motivated bounds. In this sense, OBR can serve as a natural benchmark for evaluating the accuracy of existing and future numerical methods for data transfer with respect to a given class of flux reconstruction methods and flux distance measures. In this context, we perform numerical comparisons between OBR, FCR and iFCR (a version of FCR which utilizes an iterative procedure to enhance the accuracy of FCR numerical fluxes).

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