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

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。

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

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 2nd order convergent. We prove that VOC scheme is maximum-principle-preserving, linearity-preserving and 2nd 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 2nd 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 2nd order convergent on distorted meshes for diffusion equations with discontinuous coefficients. The linear scheme which adopts VOC scheme is linearity-preserving and 2nd order convergent, and the positivity-preserving scheme is also 2nd order convergent.