三维多面体网格上扩散方程的保正格式被引量：2

A POSITIVITY-PRESERVING FINITE VOLUME SCHEME FOR DIFFUSION EQUATIONS ON POLYHEDRAL MESHES

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摘要针对三维任意(星形)多面体网格,本文构造了扩散方程的一种单元中心型非线性有限体积格式,证明了该格式具有保正性.在该格式设计中,除引入网格中心量外,还引入网格节点量和网格面中心量作为中间未知量,它们将用网格中心未知量线性组合表示,使得格式仅有网格中心未知量作为基本未知量.在节点量计算中,利用网格面上的调和平均点,设计了一种适用于三维多面体网格的局部显式加权方法.该格式适用于求解非平面的网格表面和间断扩散系数的问题.数值例子验证了它对光滑解具有二阶精度和保正性. We construct a nonlinear cell-centered finite volume scheme for diffusion equation on star-shaped polyhedral meshes and prove that it is positivity-preserving. Based on harmonic average point, we design a new locally explicit weighted method to calculate intermediate unknowns, including the vertex and face unknowns. Our scheme is applicable for distorted meshes with cell-faces being non-plane, and suitable for diffusion problems with discontinuous coefficient. Numerical examples verify the convergence and positivity of numerical solution of our scheme.

作者王帅杭旭登袁光伟

机构地区 IAPCM

出处《计算数学》 CSCD 北大核心 2015年第3期247-263,共17页 Mathematica Numerica Sinica

基金国家自然科学基金(11171036,11401034,11371066,11271054)资助项目

关键词多面体网格保正格式调和平均点扩散方程 polyhedral meshes, positivity-preserving scheme, harmonic average point, diffusion equations

分类号 O241.82 [理学—计算数学]

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1Li D Y, Chen G N. Introduction of Difference Methods for Parabolic Equations[M]. Science Press (in Chinese), Beijing, 1995.
2Shashkov M, Steinberg S. Solving diffusion equations with rough coefficients in rough grids[M]. J. Comput. Phys., 1996, 129: 383-405.
3Aavatsmark I. An introduction to multipoint flux approxmations for quadrilateral grids[J]. Com- putational Geosciences, 2002, 6: 405-432.
4Lipnikov K, Manzini G, Svyatskiy D. Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems[J]. Journal of Computational Physics, 2011, 230: 2620-2642.
5Nordbotten J M, Aavatsmark I and Eigestad G T. Monotonicity of control volume methods, Numer. Math. 2007, 106: 255-288.
6Nordbotten J M and Eigestad G T. Discretization on quadrilateral grids with improved mono- tonicity[J]. J. Comput. Phys., 2005, 203: 744-760.
7Le Potier C. Finite volume monotone scheme for highly anisotropic diffusion operators on un- structured triangular meshes[J]. C. R. Acad. Sci. Paris, Ser. I 2005, 341: 787-792.
8Lipnikov K, Shashkov M, Svyatskiy D, and Vassilevski Yu. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes[J]. J. Comput. Phys., 2007, 227: 492-512.
9Yuan G W, Sheng Z Q. Monotone finite volume schemes for diffusion equations on polygonal meshes[J]. J. Comput. Phys., 2008, 227: 6288-6312.
10袁光伟,盛志强,岳晶岩.扩散方程保正的有限体积格式[J].中国科学：数学,2012,42(9):951-970. 被引量：5