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首页> 外文期刊>Journal of Computational Physics >Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations
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Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations

机译：基于优化的重映射和传输：保留特征离散化的分而治之策略

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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.
机译：本文考察了优化和控制思想在特征保留数值方法的制定中的应用，特别强调了单个标量的保守和保留边界重映射（约束插值）和传输（对流）。我们提出了一种用于保留物理性质的通用优化框架，并将其专门用于基于质量的通用优化重映射（OBR）。后者将重映射转换为二次方程序，其最佳解决方案在受到线性不等式约束的情况下，将与适当目标量的距离最小化。精确的质量更新算子的近似值定义了目标数量，该目标数量可提供新质量的最佳精度，而无需考虑任何物理约束（例如质量守恒或局部边界）。后者由线性不等式系统强制执行。这样，通用OBR公式将准确性考虑与物理属性的执行分开。我们将继续展示通用OBR配方如何产生重新引入的最近引入的通量可变通量目标（FVFT）[1]和质量可变质量目标（MVMT）[2]配方，然后对其形式进行正式检查关系。使用中间通量可变质量目标（FVMT）公式，我们显示了FVFT和MVMT最优解的等效性。为了强调通用OBR公式的范围和多功能性，我们引入了适应性目标的概念，即反映局部解决方案属性，扩展FVFT和MVMT以在球体上重新映射的目标量，并使用OBR来制定适应性，保守性和约束性保留用于笛卡尔坐标系和经纬度坐标系的基于优化的传输算法。在二维网格上选择了具有代表性的数值示例，证明了我们方法的计算特性。

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