单调有限差分方案用于各向异性扩散问题的非负方向分裂
Monotone finite difference schemes for anisotropic diffusion problems via nonnegative directional splittings
AI总结：
本文研究了各向异性扩散算子的非负方向分裂，构建了单调有限差分方案，证明了在满足一定条件时可构造单调方案，且给出了 stencil 大小的上界。
摘要：
非负方向分裂被用来研究各向异性扩散算子在散度形式下的性质。在任意内部点的邻域内建立了非负方向分裂的条件。该结果用于构建各向异性扩散算子的边界值问题的单调有限差分方案。证明了当底层扩散矩阵在物理域的闭合上连续，对称且在域内均匀正定，网格间距足够小，有限差分Stencil大小足够大时，可以构造单调方案。得到了Stencil大小的上界，其完全由扩散矩阵决定。一般来说，扩散矩阵越各向异性，所需Stencil越大。例外情况是严格对角占优的扩散矩阵，此时三阶Stencil足以构造单调有限差分方案。数值例子用于验证理论结果。
英文摘要：
Nonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The result is used to construct monotone finite difference schemes for the boundary value problem of anisotropic diffusion operators. It is shown that such a monotone scheme can be constructed if the underlying diffusion matrix is continuous on the closure of the physical domain and symmetric and uniformly positive definite on the domain, the mesh spacing is sufficiently small, and the size of finite difference stencil is sufficiently large. An upper bound for the stencil size is obtained, which is determined completely by the diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the larger stencil is required. An exception is the situation with a strictly diagonally dominant diffusion matrix where a three-by-three stencil is sufficient for the construction of a monotone finite difference scheme. Numerical examples are presented to illustrate the theoretical findings.
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