Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations

This paper examines the application of optimization and control ideas to the formulation of feature-preserving numerical methods, with particular emphasis on the conservative and bound-preserving remap (constrained interpolation) and transport (advection) of a single scalar quantity. We present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization-based remap (OBR) of mass density. The latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity, subject to a system of linear inequality constraints. The approximation of an exact mass update operator defines the target quantity, which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds. The latter are enforced by the system of linear inequalities. In so doing, the generic OBR formulation separates accuracy considerations from the enforcement of physical properties. We proceed to show how the generic OBR formulation yields the recently introduced flux-variable flux-target (FVFT) [1] and mass-variable mass-target (MVMT) [2] formulations of remap and then follow with a formal examination of their relationship. Using an intermediate flux-variable mass-target (FVMT) formulation we show the equivalence of FVFT and MVMT optimal solutions. To underscore the scope and the versatility of the generic OBR formulation we introduce the notion of adaptable targets, i.e., target quantities that reflect local solution properties, extend FVFT and MVMT to remap on the sphere, and use OBR to formulate adaptable, conservative and bound-preserving optimization-based transport algorithms for Cartesian and latitude/longitude coordinate systems. A selection of representative numerical examples on two-dimensional grids demonstrates the computational properties of our approach.

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