Abstract In the construction of cell-centered finite volume schemes for general diffusion equations with discontinuous coefficients on distorted meshes, auxiliary unknowns are usually introduced to resolve discontinuities across cell-edges. In general, it is difficult to express unconditionally auxiliary unknowns as a convex combination of primary unknowns around. As we know, all existing methods have to impose certain restrictive conditions on cell-distortion and coefficient discontinuities, especially when designing discrete schemes satisfying the maximum principle. In this paper, we present a nonlinear convex combination method, in which each vertex unknown is eliminated by a nonlinear convex combination of two cell-centered unknowns, which are respectively the maximum and minimum of those cell-centered unknowns around the vertex. As two application examples we present two cell-centered schemes satisfying the maximum principle based on the nonlinear convex combination. An analysis for the discrete flux shows that our scheme can exactly recover the linear solution. Numerical results are presented to show the accuracy of the resulting schemes and verify the discrete maximum principle.

中文翻译：

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满足极大值原理的有限体积方案构造中的非线性凸组合

摘要 在扭曲网格上具有不连续系数的一般扩散方程的以单元为中心的有限体积方案的构造中，通常引入辅助未知数来解决跨单元边缘的不连续性。一般来说，很难将无条件辅助未知数表达为周围主要未知数的凸组合。众所周知，所有现有方法都必须对单元失真和系数不连续性施加一定的限制条件，尤其是在设计满足最大值原则的离散方案时。在本文中，我们提出了一种非线性凸组合方法，其中每个顶点未知由两个以单元为中心的未知数的非线性凸组合消除，这两个未知数分别是顶点周围那些以单元为中心的未知数的最大值和最小值。作为两个应用示例，我们提出了两种基于非线性凸组合的满足最大值原理的以单元为中心的方案。对离散通量的分析表明，我们的方案可以准确地恢复线性解。数值结果显示了所得方案的准确性并验证了离散最大值原理。