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 relate the recently introduced flux-variable fluxtarget (FVFT) [1] and mass-variable mass-target (MVMT) [2] optimization-based remap (OBR) to this framework. Both cast remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity, subject to a system of linear inequality constraints. An 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, OBR separates accuracy considerations from the enforcement of the physical properties. We follow with a formal examination of the relationship between the FVFT and MVMT formulations. Using an intermediate flux-variable mass-target (FVMT) formulation we show the equivalence of their optimal solutions. To underscore the scope and the versatility of the OBR approach 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, both in R and on the sphere. A selection of representative numerical examples demonstrates the computational properties of our approach.