二维扩散方程的局部二阶线性保极值节点计算方法及应用  

Applications of Local Maximum-principle-preserving Linear Vertex Scheme of Second Order Convergence for Two-dimensional Diffusion Equations

作　　者：张淑娴 杭旭登[1,3] ZHANG Shuxian;HANG Xudeng(Institute of Applied Physics and Computational Mathematics,Beijing 100094,China;Graduate School of China Academy of Engineering Physics,Beijing 100088,China;Laboratory of Computational Physics,Beijing 100088,China)

机构地区：[1]北京应用物理与计算数学研究所,北京100094 [2]中国工程物理研究院研究生院,北京100088 [3]计算物理实验室,北京100088

出　　处：《计算物理》2022年第1期17-32,共16页Chinese Journal of Computational Physics

基　　金：国家自然科学基金(11971069);中物院创新发展基金(CX20210021);计算物理重点实验室基金(6142A05180203)资助项目。

摘　　要：对一般四边形网格设计一种优化的节点控制体,并构造了一种扩散方程的保极值二阶收敛的局部线性节点计算格式(优化控制体节点格式,VOC格式)。在网格不出现异常节点的情况下,证明VOC格式是保极值、线性精确和二阶收敛的。而且在均匀的矩形网格上,修正的逆距离加权格式与VOC格式等价,从而对间断系数问题也是局部二阶收敛的。VOC格式可以用于单元中心型线性扩散格式和保正格式的节点值计算。数值算例表明对扭曲网格上的间断系数问题,VOC格式是二阶收敛的。采用VOC格式计算节点值的线性九点格式具有线性精确性和二阶收敛性,采用VOC格式的保正格式也具有二阶收敛性。We construct a kind of optimized control volume for general quadrilateral meshes. Based on the control volume, we devise a local linear vertex scheme(Vertex-scheme on Optimized Control volume, VOC), which is maximum-principle-preserving, and 2 nd order convergent. We prove that VOC scheme is maximum-principle-preserving, linearity-preserving and 2 nd order convergent if no exceptional vertex exists. On uniform rectangle meshes, we prove that the modified inverse distance weight(MIDW) scheme is approximately VOC scheme, and they are all 2 nd order convergent. VOC scheme can be used to construct linear cell-centered diffusion schemes and positivity-preserving diffusion schemes. Numerical experiments verify that the scheme is 2 nd order convergent on distorted meshes for diffusion equations with discontinuous coefficients. The linear scheme which adopts VOC scheme is linearity-preserving and 2 nd order convergent, and the positivity-preserving scheme is also 2 nd order convergent.

关 键 词：节点格式 修正的逆距离加权格式 有限体积方法 扩散方程 保极值 

分 类 号：O242[理学—计算数学]