A family of linearity-preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids
A family of linearity-preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids
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DOI：
10.1016/j.cma.2013.08.006
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发表时间：
2013-12
影响因子：
7.2
通讯作者：
Weiwei Sun;Jiming Wu;Xiaoping Zhang
中科院分区：
文献类型：
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作者：
Weiwei Sun;Jiming Wu;Xiaoping Zhang
针对任意平面面体网格上的各向异性扩散问题，提出了一族以单元为中心的有限体积格式。方案的推导是在一般框架下通过一定的线性保持方法完成的。该算法的关键是只使用位于网格界面的调和平均点来定义辅助未知数，这不仅使得辅助未知数的内插过程简单且保持正性，而且减少了格式的模版。最终的方案是以单元为中心，在结构化的六面体网格上使用25个点的小模板。此外，该格式满足局部守恒条件，能够准确地处理间断，并允许进行简单的稳定性分析。在扩散张量为各向异性和不连续的情况下，在一般的扭曲网格上，数值模拟得到了L 2范数的二阶精度和H1范数的一阶精度。
A family of cell-centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets. The derivation of the schemes is done under a general framework through a certain linearity-preserving approach. The key ingredient of our algorithm is to employ solely the so-called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns, which not only makes the interpolation procedure for auxiliary unknowns simple and positivity-preserving, but also reduces the stencil of the schemes. The final schemes are cell-centered with a small stencil of 25-point on the structured hexahedral grids. Moreover, the schemes satisfy the local conservation condition, treat discontinuity exactly and allow for a simple stability analysis. A second-order accuracy in the L 2 norm and a first-order accuracy in the H 1 norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous.