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

In this report we use optimization ideas to develop and study conservative, ∗Corresponding author Email addresses: [email protected] (Pavel Bochev), [email protected] (Denis Ridzal), [email protected] (Guglielmo Scovazzi), [email protected] (Mikhail Shashkov ) URL: http://www.sandia.gov/~pbboche/ (Pavel Bochev), http://www.sandia.gov/~dridzal/ (Denis Ridzal), http://www.sandia.gov/~gscovaz/ (Guglielmo Scovazzi), http://cnls.lanl.gov/~shashkov/ (Mikhail Shashkov ) Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC0494AL85000 Preprint submitted to Journal of Computational Physics June 14, 2010 bound-preserving algorithms for remap of a scalar conserved quantity (mass) between close meshes having the same connectivity. We formulate remap as an inequality-constrained optimization problem in which the objective is to minimize the distance between given high-order target mass fluxes and the mass exchanges between neighboring cells, subject to constraints derived from physically motivated bounds on the associated primitive variable (density). In doing so, we separate accuracy considerations, handled by the objective functional, from the enforcement of physical bounds, handled by the constraints. A typical high-order remap algorithm enforces bounds by a direct manipulation of the reconstruction process using slope limiters, which is standard practice in numerical algorithms for the solution of advection problems. In contrast, the new optimization-based remap (OBR) finds the most accurate, with respect to the selected distance measure, remapped quantity from a feasible set defined by physical bounds. As a result, the OBR formulation can be easily applied to unstructured grids and grids comprising of arbitrary cell shapes. Moreover, under some additional assumptions on the grid motion, but not on the cell types, we prove that the OBR algorithm is linearity preserving in one, two and three dimensions. The report also examines connections between OBR and the recently proposed flux-corrected remap (FCR) [1]. We show that FCR can be interpreted as a solution procedure for a modified version of the OBR problem (M-OBR) in which the same objective is minimized over a subset of the OBR feasible set. The modified feasible set is derived by considering a “worstcase” scenario that replaces the original constraints by a simplified set of box constraints. The simplified constraint set decouples M-OBR into a series of one-dimensional minimization problems that can be solved independently from each other. The resulting M-OBR solution coincides with the FCR solution. It thus follows that OBR is always at least as accurate as FCR. The report concludes with a numerical study of the OBR and FCR formulations in one dimension. We compare qualitative properties, such as shape preservation, and estimate the rates of convergence using remap on a series of smooth and “hourglass” grid cycles. The study confirms that the larger feasible set of OBR delivers increased accuracy compared to FCR.