任意多边形上非定常扩散方程的单元中心型有限体积格式

Cell-centered Finite Volume Scheme for Evolutionary Diffusion Equations on Arbitrary Polygonal Meshes

针对二维非定常扩散方程,构造适用于任意多边形网格的单元中心型有限体积格式。采用向后欧拉格式进行时间离散,空间上在离散扩散算子时,利用网格顶点作为辅助插值点,通过求解一个欠定方程组将辅助插值点信息替换成网格单元中心点信息,最终得到只含单元中心未知量的离散格式。该格式既满足局部守恒条件,又满足线性精确准则。在几类多边形网格上进行数值实验,分别考虑扩散系数是连续和间断的情况,发现新格式均可达到二阶收敛。其数值表现显著优于算数平均加权和逆距离加权的九点格式,与双线性插值的加权方式结果相近,并且克服了双线性插值加权方式不适用于三角形网格的弊端。数值算例表明新格式求解非线性扩散方程仍然可以达到二阶收敛。

A cell-centered finite volume scheme for the 2D evolutionary diffusion equation on arbitrary polygonal meshes is constructed. We apply the backward Euler scheme to discrete the time derivative term, and employ the vertex unknowns as auxiliary ones to discrete the diffusion operator, by solving an underdetermined linear system of equations, vertex unknowns can be expressed by a linear combination of the central unknowns, which finally results in a cell-centered scheme. The proposed scheme maintains the local conservation and the linearity preserving properties. Considering the continuous and discontinuous diffusion coefficients respectively, several numerical experiments on different kinds of polygonal meshes show that second-order convergence rate can be obtained. Its numerical performance is significantly better than the nine point scheme with arithmetic average weighting and inverse distance weighting,and is similar to the weighting method of bilinear interpolation, it overcomes the disadvantage that bilinear interpolation is not suitable for triangular meshes. Besides, the numerical results also implies that proposed scheme can still achieve second-order convergence for solving nonlinear diffusion equations.