High-resolution conservative algorithms for advection in incompressible flow
SIAM Journal on Numerical Analysis
A monotone finite element scheme for convection-diffusion equations
Mathematics of Computation
Incremental remapping as a transport&slash;advection algorithm
Journal of Computational Physics
The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods
SIAM Journal on Numerical Analysis
Second-order sign-preserving conservative interpolation (remapping) on general grids
Journal of Computational Physics
An efficient linearity-and-bound-preserving remapping method
Journal of Computational Physics
SIAM Journal on Numerical Analysis
The repair paradigm: New algorithms and applications to compressible flow
Journal of Computational Physics
Journal of Computational Physics
Non-negative mixed finite element formulations for a tensorial diffusion equation
Journal of Computational Physics
A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid
Journal of Computational Physics
Local adaptive mesh refinement for shock hydrodynamics
Journal of Computational Physics
An Optimization-Based Approach for the Design of PDE Solution Algorithms
SIAM Journal on Numerical Analysis
On maximum-principle-satisfying high order schemes for scalar conservation laws
Journal of Computational Physics
A class of deformational flow test cases for linear transport problems on the sphere
Journal of Computational Physics
Journal of Computational Physics
A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Additive operator decomposition and optimization–based reconnection with applications
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
Optimization---Based modeling with applications to transport: part 1. abstract formulation
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
Optimization-Based modeling with applications to transport: part 2. the optimization algorithm
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
Optimization-Based modeling with applications to transport: part 3. computational studies
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
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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.