A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids
A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids
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DOI：
10.1016/j.jcp.2017.04.070
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发表时间：
2017-09
期刊：
影响因子：
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通讯作者：
Xiaoping Zhang;Shuai Su;Jiming Wu
中科院分区：
文献类型：
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作者：
Xiaoping Zhang;Shuai Su;Jiming Wu
对任意多边形网格上的各向异性扩散问题提出了一种新的保正有限体积格式。该方案具有点中心，边中点和细胞中心的未知数。以顶点为中心的未知量是主要的，并且具有与它们相关联的有限体积方程。将边中点和单元中心未知量作为辅助未知量，用主未知量进行插值，使最终格式成为纯顶点中心格式。与大多数现有的正性保持计划，该计划的建设是基于一个特殊的非线性两点通量近似，有一个固定的模板，不需要凸分解的共法。为了有效地求解由非线性格式产生的非线性方程组，讨论了Picard方法及其安德森加速。数值实验表明，在严重变形网格上，该方法对非均匀各向异性问题的解具有二阶精度和良好的正性。安德森加速的高效率也表现在非线性迭代次数的减少上。此外，所提出的计划没有所谓的数值热障碍问题所遭受的大多数现有的细胞中心和混合计划。但如果解非常接近机器精度且网格变形非常严重，则需要进一步改进。
We suggest a new positivity-preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids. The scheme has vertex-centered, edge-midpoint and cell-centered unknowns. The vertex-centered unknowns are primary and have finite volume equations associated with them. The edge-midpoint and cell-centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. Unlike most existing positivity-preserving schemes, the construction of the scheme is based on a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal. In order to solve efficiently the nonlinear systems resulting from the nonlinear scheme, Picard method and its Anderson acceleration are discussed. Numerical experiments demonstrate the second-order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the Anderson acceleration is also shown on reduction of the number of nonlinear iterations. Moreover, the proposed scheme does not have the so-called numerical heat-barrier issue suffered by most existing cell-centered and hybrid schemes. However, further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe.