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Formulation, analysis and numerical study of an optimization-based conservative interpolation (remap) of scalar fields for arbitrary Lagrangian-Eulerian methods

Bochev, Pavel; Ridzal, Denis R.; Scovazzi, Guglielmo S.; Shashkov, Mikhail

We develop and study the high-order conservative and monotone optimization-based remap (OBR) of a scalar conserved quantity (mass) between two close meshes with the same connectivity. The key idea is to phrase 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 (density). In so doing, we separate accuracy considerations, handled by the objective functional, from the enforcement of physical bounds, handled by the constraints. The resulting OBR formulation is applicable to general, unstructured, heterogeneous grids. Under some weak requirements on grid proximity, but not on the cell types, we prove that the OBR algorithm is linearity preserving in one, two and three dimensions. The paper also examines connections between the OBR and the recently proposed flux-corrected remap (FCR), Liska et al. [1]. We show that the FCR solution coincides with the solution of a modified version of OBR (M-OBR), which has the same objective but a simpler set of box constraints derived by using a "worst-case" scenario. Because M-OBR (FCR) has a smaller feasible set, preservation of linearity may be lost and accuracy may suffer for some grid configurations. Our numerical studies confirm this, and show that OBR delivers significant increases in robustness and accuracy. Preliminary efficiency studies of OBR reveal that it is only a factor of 2.1 slower than FCR, but admits 1.5 times larger time steps. © 2011 Elsevier Inc.