We construct a new scheme whose primary unknowns include both cell-centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient. First, two linear fluxes are given on both sides of the cell edge, respectively. In order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous coefficient, in addition to cell-centered unknowns, we also introduce cell-edge unknowns on discontinuous line as basic unknowns. Second, the conservative flux is constructed by using nonlinear weighted combination of these two linear fluxes. For the cell-edge unknowns on discontinuous line, we add an equation by using the continuity of normal flux. Compared to the classical cell-centered nonlinear finite volume scheme, the introduction of cell-edge unknowns on discontinuous edge is the key point for our scheme to solve diffusion equations with discontinuous coefficient. Then we prove that the scheme satisfies the discrete maximum principle. Based on this, the existence of a solution for the scheme is also obtained. Numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes, preserves discrete maximum principle, and is superior to the existing scheme (in Sheng and Yuan (2011)) preserving the maximum principle on dealing with the problems with strong anisotropic discontinuous coefficient on distorted meshes.

我们构造了一个新方案，其主要未知数包括不连续线上的以单元为中心的未知数和边缘未知数，以求解具有不连续系数的扩散方程。首先，在单元边缘的两侧分别给出两个线性通量。为了解决现有方案保留最大原理来解决具有不连续系数的扩散问题的缺陷，除了以单元为中心的未知量之外，我们还以不连续线为基础来引入不连续单元上的单元边缘未知量。其次，通过使用这两个线性通量的非线性加权组合来构造保守通量。对于不连续线上的细胞边缘未知数，我们使用法向通量的连续性添加一个方程式。与经典的以细胞为中心的非线性有限体积方案相比，在不连续边上引入单元边缘未知数是我们求解具有不连续系数的扩散方程的方案的关键。然后证明了该方案满足离散最大原理。基于此，还获得了该方案的解决方案的存在。数值结果表明，我们的方案在随机网格上的求解获得了几乎二阶的精度，保留了离散的最大原理，并且优于现有方案（Sheng和Yuan（2011）），保留了解决问题的最大原理。在变形网格上具有很强的各向异性不连续系数。基于此，还获得了该方案的解决方案的存在。数值结果表明，我们的方案在随机网格上的求解获得了几乎二阶的精度，保留了离散的最大原理，并且优于现有方案（Sheng和Yuan（2011）），保留了解决问题的最大原理。在变形网格上具有很强的各向异性不连续系数。基于此，还获得了该方案的解决方案的存在。数值结果表明，我们的方案在随机网格上的求解获得了几乎二阶的精度，保留了离散的最大原理，并且优于现有方案（Sheng和Yuan（2011）），保留了解决问题的最大原理。在变形网格上具有很强的各向异性不连续系数。