三维非匹配网格上的扩散方程数值求解

计算物理 ›› 2017, Vol. 34 ›› Issue (1): 19-28.

三维非匹配网格上的扩散方程数值求解

郭少冬, 章明宇, 周海兵, 熊俊, 张树道

北京应用物理与计算数学研究所, 北京 100094

收稿日期:2015-11-17 修回日期:2016-03-07 出版日期:2017-01-25 发布日期:2017-01-25
作者简介:郭少冬(1982-),男,博士,副研究员,从事辐射流体力学数值模拟研究,E-mail:guo_shaodong@iapcm.ac.cn
基金资助:
国家自然科学基金项目（11205016，11472060，11372050）；中国工程物理研究院科学技术发展基金项目（2014B0202031，2013A0101004）资助

Solving Diffusion Equation on Three-Dimensional Non-Conformal Mesh

GUO Shaodong, ZHANG Mingyu, ZHOU Haibing, XIONG Jun, ZHANG Shudao

Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

Received:2015-11-17 Revised:2016-03-07 Online:2017-01-25 Published:2017-01-25

摘要/Abstract

摘要： 将子网格剖分的支撑算子方法，拓展应用于三维非匹配网格上的扩散方程求解.算例表明该方法在正交非匹配网格上能够精确获得线性解；在一般非匹配网格上可以达到二阶精度；在求解曲面网格和节点不共面网格时，精度比平面近似的方法要高，也可以达到2阶精度，同时也适合求解含有物质界面的混合介质网格.

关键词: 扩散方程, 三维非匹配网格, 支撑算子格式, 子网格剖分

Abstract: Sub-division method based on support operator is used to solve diffusion equation with three-dimensional non-conformal mesh and non-planar mesh.Numerical experiments show that the method is second-order accurate on general non-conformal mesh.For curved-face mesh and non-planar-face mesh, the method is more accurate than traditional plane-approximation method.For non-conformal orthogonal mesh, the method can obtain accurate solutions of linear problems.

Key words: diffusion equation, three-dimensional non-conformal mesh, support operator method, sub-division

中图分类号:

O532

郭少冬, 章明宇, 周海兵, 熊俊, 张树道. 三维非匹配网格上的扩散方程数值求解[J]. 计算物理, 2017, 34(1): 19-28.

GUO Shaodong, ZHANG Mingyu, ZHOU Haibing, XIONG Jun, ZHANG Shudao. Solving Diffusion Equation on Three-Dimensional Non-Conformal Mesh[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2017, 34(1): 19-28.

参考文献

[1] 李德元,水鸿寿,汤敏君. 关于非矩形网格上的二维抛物型方程的差分格式[J]. 数值计算与计算机应用, 1980, 4:217-224.
[2] KERSHAW D S. Differencing of the diffusion equation in Lagrangian hydrodynamics codes[J]. Journal of Computational Physics, 1981, 39:375-395.
[3] MOREL J E, DENDY J E, HALL M, et al. A cell-centered Lagrangian mesh differencing scheme[J]. Journal of Computational Physics, 1992, 103:286-299.
[4] SHASHKOV M, STEINBERG S. Solving diffusion equations with rough coefficients in rough grids[J]. Journal of Computational Physics, 1996, 129:383-405.
[5] MOREL J E, ROBERTS R M, SHASHKOV M. A local support-operators diffusion discretization scheme for quadrilateral r-z meshes[J]. Journal of Computational Physics, 1998, 144:17-51.
[6] 张瑗,李寿佛. 在不规则四边形网格上逼近扩散算子的五点及九点差分格式的测试[J].湘潭大学自然科学学报,2002,24(4):12-17.
[7] SHENG Z Q, YUAN G W. A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes[J]. SIAM Journal on Scientific Computing, 2008, 30(3):1341-1361.
[8] 袁光伟. 扭曲网格上扩散方程九点格式的构造与分析. 计算物理实验室年报,2005:530-575.
[9] 聂存云,舒适,杭旭登,成娟. 柱坐标系下辐射热传导方程的SFVE格式[J].中国核科学技术进展报告,2009,(1):92-106.
[10] LIPNIKOV K, MOREL J, SHASHKOV M. Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes[J]. Journal of Computational Physics, 2004, 199:589-597.
[11] CHANG L N, YUAN G W. Cell-centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198:1638-1646.
[12] BREZZI F, LIPNIKOV K, SHASHKOV M. Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes[J]. SIAM J Numer Anal, 2005, 43(5):1872-1896.
[13] BREZZI F, LIPNIKOV K, SHASHKOV M. Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces[J]. Mathematical Models and Methods in Applied Science, 2006, 16(2):275-297.
[14] LIPNIKOV K, SHASHKOV M, SVYATSKIY D. The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes[J]. Journal of Computational Physics, 2006, 211:473-491.
[15] GARIMELLA R V, LIPNIKOV K. Solution of the diffusion equation in multi-material domains by sub-division of elements along reconstructed interfaces[J]. International Journal for Numerical Methods in Fluids, 2011, 65(11-12):1423-1437.