保双线性代数通量修正
Bilinearity Preserving Algebraic Flux Correction
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- Indian Institute of Technology Gandhinagar（印度甘地纳加尔理工学院）
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中文总结 AI 辅助
本文提出保双线性代数通量修正限制器，在轴对齐网格上精确再现双线性解，优于线性保持限制器，并给出参数显式值及网格条件。
中文摘要 AI 辅助
对流-扩散-反应方程的代数通量修正格式依赖于线性保持以避免不必要的数值扩散：当离散解为仿射函数时，限制器不激活。在四边形和六面体网格上，这并不充分，因为有限元空间还包含在轴对齐网格上被精确再现的双线性函数。我们证明，当且仅当将每个邻居提升其坐标差之积后，中心节点位于其邻居的凸包内时，网格补丁才允许双线性保持，并且当单元边平行于坐标轴时，我们推导出此类补丁上限制器参数的尖锐显式值。我们进一步刻画了映射双线性有限元空间精确再现双线性函数的网格，从而确立了任何限制器在一般四边形上所能达到的基本限制。数值实验证实了该理论：在轴对齐网格上，所提出的限制器将双线性解再现到非线性迭代的精度，而线性保持限制器则不能。敏感性研究表明，双线性保持所需的参数接近此处考虑的定点迭代可求解非线性问题的范围的上端。
英文摘要
Algebraic flux correction schemes for convection--diffusion--reaction equations rely on linearity preservation to avoid unnecessary artificial diffusion: the limiter is inactive whenever the discrete solution is affine. On quadrilateral and hexahedral meshes, this is insufficient, since the finite element space also contains bilinear functions that are reproduced exactly on axis-aligned meshes. We show that a mesh patch admits bilinearity preservation if and only if the central node lies in the convex hull of its neighbours after lifting each neighbour by the product of its coordinate differences, and we derive a sharp explicit value for the limiter parameter on such patches when the cell edges are parallel to the coordinate axes. We further characterize the meshes on which a mapped bilinear finite element space reproduces a bilinear function exactly, thereby establishing a fundamental limitation on what any limiter can achieve on general quadrilaterals. Numerical experiments confirm the theory: on axis-aligned grids, the proposed limiter reproduces a bilinear solution to the accuracy of the nonlinear iteration, whereas the linearity-preserving limiter does not. A sensitivity study shows that the parameter required for bilinearity preservation lies close to the upper end of the range for which the nonlinear problem can be solved by the fixed-point iteration considered here.
