We propose a new second-order accurate finite volume (FV) scheme for diffusion equations with discontinuous coefficients on unstructured meshes. This scheme is based on a novel least squares reconstruction with a compact neighbourhood that computes the cell-centered diffusion fluxes from the normal diffusion flux at cell faces. We propose two variants of the scheme rooted in the concept of “alpha damping” (AD), which are both linearly exact on arbitrary mesh topologies even when the tangential diffusive fluxes are discontinuous. The first variant, referred to as AD-G (Alpha Damping-Gradient), effects the discretisation of the normal derivative while the second, referred to as AD-F (Alpha Damping-Flux), involves discretisation of the normal fluxes. It is shown that harmonic averaging for diffusion coefficient at the faces is essential for the accuracy of the solution with the AD-G scheme but this condition is not necessary for the AD-F scheme. Numerical experiments demonstrate that the proposed FV schemes estimate the solution and fluxes to second and first order accuracy respectively for discontinuous diffusion problems on generic polygonal meshes. Studies also show that these schemes are discrete extremum preserving (DEP) and can be implemented with relative ease for diffusion problems in existing legacy codes.

对于非结构网格上具有不连续系数的扩散方程，我们提出了一种新的二阶精确有限体积（FV）方案。该方案基于具有紧凑邻域的新型最小二乘重构，该紧凑邻域可以根据单元表面的法向扩散通量来计算以单元为中心的扩散通量。我们提出了基于“α阻尼”（AD）概念的方案的两个变体，即使切向扩散通量不连续，它们在任意网格拓扑上都线性精确。第一个变量称为AD-G（Alpha阻尼梯度），影响正态导数的离散化，而第二个变量称为AD-F（Alpha阻尼通量），涉及法线通量的离散化。结果表明，对于AD-G方案，精度对于面处扩散系数的谐波平均对于解决方案的准确性至关重要，但是对于AD-F方案，此条件不是必需的。数值实验表明，对于通用多边形网格上的不连续扩散问题，所提出的FV方案分别将解和通量估计为二阶和一阶精度。研究还表明，这些方案是离散的极值保留（DEP），可以相对容易地实现，以解决现有遗留代码中的扩散问题。数值实验表明，对于通用多边形网格上的不连续扩散问题，所提出的FV方案分别将解和通量估计为二阶和一阶精度。研究还表明，这些方案是离散的极值保留（DEP），可以相对容易地实现，以解决现有遗留代码中的扩散问题。数值实验表明，对于通用多边形网格上的不连续扩散问题，所提出的FV方案分别将解和通量估计为二阶和一阶精度。研究还表明，这些方案是离散的极值保留（DEP），可以相对容易地实现，以解决现有遗留代码中的扩散问题。