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\begin{cabstract}
本文研究各向异性扩散方程的数值求解. 在多介质辐射流体力学问题中, 扩散系数通常是间断的、各向异性的张量. 本文在任意变形的多面体网格上, 对含有任意各向异性扩散系数的扩散方程构造高效、鲁棒的有限体积格式. 单元中心型格式广泛应用于多介质辐射流体, 本文根据线性精确的思想, 推导一类二阶精度的单元中心型有限体积格式, 探索构造三维扩散格式的新途径. 我们首先基于线性精确准则将二维九点格式推广为三维菱形格式. 菱形格式的名称源于通量近似所采用的菱形模板, 网格面上的法向流表示为面两侧的单元未知量和面上的顶点未知量的组合.
然后, 我们将顶点未知量视为辅助未知量, 由周围单元中心未知量插值消去.
我们使用线性精确准则, 构造三维多面体网格上的两种线性精确的顶点插值算法, 第一种顶点插值算法使用多点通量近似结合取极限过程获得, 第二种顶点插值算法使用最小二乘技术结合图搜索算法获得. 两种顶点插值算法在任意多面体网格和任意各向异性扩散张量下都保持线性精确性质.

本文还研究了各向异性扩散方程的单元中心型有限体积格式在多孔介质两相流模型的应用. 数学模型包括压力的扩散方程和饱和度的非线性双曲方程. 这里, 压力的扩散方程是一个含有各向异性间断系数的二阶椭圆方程. 本文在任意非结构网格上, 对含有任意各向异性扩散系数的压力扩散方程, 发展高效、鲁棒的有限体积格式. 单元中心型格式广泛应用于多介质辐射流体, 本文根据线性精确的思想, 推导一类二阶精度的单元中心型有限体积格式, 并将其应用于多孔介质两相流数值模拟. 结合双曲守恒律中广泛使用的二阶单调MUSCL格式求解饱和度方程, 进而形成格式可扩展的两相流模拟器. 此外, 本文的研究成果可推广应用于例如金属铸造等两相驱替的工程问题.
\end{cabstract}

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This paper studies the numerical solution for anisotropic diffusion equations. In multi-material radiative hydrodynamics problems, the diffusion coefficient is usually a discontinuous and anisotropic tensor. In this paper, an efficient and robust finite volume scheme for diffusion equations with arbitrary anisotropic diffusion coefficients is constructed on arbitrarily polyhedral meshes. The cell-centered scheme is widely used in multi-material radiative hydrodynamics. Based on the idea of linearity-preserving, a class of second-order accurate cell-centered finite volume schemes is derived in this paper, exploring a new approach to constructing three-dimensional diffusion schemes. We first extend the two-dimensional nine-point scheme to a three-dimensional diamond scheme based on the linearity-preserving criterion. The name of the diamond scheme comes from the diamond-shaped stencil used in the flux approximation, and the normal flux on the mesh face is expressed as a combination of the cell unknowns on both sides of the face and the vertex unknowns on the face. Then, we treat the vertex unknowns as auxiliary unknowns and eliminate them by interpolating from the surrounding cell-centered unknowns. We use the linearity-preserving criterion to construct two linearity-preserving vertex interpolation algorithms on three-dimensional polyhedral meshes. The first vertex interpolation algorithm is obtained by combining multi-point flux approximation with a limiting process, and the second vertex interpolation algorithm is obtained by combining least squares techniques with graph search algorithms. Both vertex interpolation algorithms maintain linearity-preserving on arbitrary polyhedral meshes and for arbitrary anisotropic diffusion tensors.

In this paper, we investigate the application of the cell-centered finite volume scheme for the anisotropic diffusion equation in the two-phase flow model in porous media. The mathematical model comprises a diffusion equation for the pressure and a nonlinear hyperbolic equation for the saturation. Here, the diffusion equation for the pressure is a second-order elliptic equation with an anisotropic and eventually discontinuous diffusion coefficient. We develop efficient and robust finite volume schemes for diffusion equations with arbitrary anisotropic coefficients on arbitrary unstructured grids. In this paper, based on the idea of linearity-preserving, a family of second-order cell-centered finite volume scheme will be derived and applied to the numerical simulation of two-phase flow in porous media. The saturation equation is solved using the second-order monotone MUSCL schemes which were widely used in hyperbolic conservation law, and then a two-phase flow simulator with extensible scheme is formed. In addition, the research results of this paper can be applied to the engineering problems of two-phase displacement such as metal casting.
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