非定常扩散方程基于调和平均点插值的有限体积格式

Finite volume schemes of evolutionary diffusion equations based on harmonic averaging interpolation

本文研究非定常扩散方程适用于扭曲和非结构网格的单元中心型的有限体积格式.在网格边上离散法向流时,选取当前网格边及与其相邻网格边上的调和平均点作为辅助插值点,通过它们与单元中心点不同的组合形式给出4类法向流的离散近似,最后通过调和平均点的两点插值算法,将其替换成相邻单元的中心未知量,进而建立4种单元中心型有限体积格式.时间导数项采用向后Euler格式进行离散.该格式具有模板小、易实现的优点,满足局部守恒和二阶收敛的特性.在一定网格假设前提下,理论上证明了算法的稳定性和收敛性.数值上考虑扩散系数是连续的、间断的、各向异性的甚至依赖于未知量是非线性的等情形,分别在非结构三角形、四边形和多边形网格上进行求解.结果表明,前两种算法对不同网格不同类型扩散系数问题上的鲁棒性更好,L^(2)误差均可达到二阶收敛,H^(1)误差接近一阶甚至高于一阶收敛;后两种算法对网格的依赖性更强.

This paper focuses on developing cell-centered finite volume schemes on arbitrary distorted and unstructured meshes for evolutionary diffusion equations.On each mesh cell,by using the cell center and three harmonic averaging points on cell boundaries as the interpolation points,we approximate the normal flux in four different ways.To make these finite volume schemes cell-centered,we eliminate the intermediate variables in the flux expression by the harmonic averaging interpolation procedure.We discretize the time derivative term by the backward Euler scheme.Under some proper assumptions on meshes,we prove the stability and error estimate of the scheme theoretically.Considering the continuous,discontinuous,and heterogeneous diffusion tensors,even for the nonlinear diffusion equations,we construct four numerical experiments to investigate the convergence and robustness.Several different unstructured triangular,quadrilateral,and polygonal meshes are adopted.The numerical results imply that the former two schemes are robust for any kind of mesh and any kind of diffusion tensor,and they can obtain second-order convergence in L^(2)norm and first-order convergence in H^(1) norm;but the latter two schemes are more dependent on meshes.