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\begin{cabstract}
本文研究多介质辐射流体力学问题中扩散方程的数值求解. 多介质问题中的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. 首先将线性精确的思想用于构造三维多面体网格上的节点插值算法, 这将是首个适用于三维大变形网格的二阶精度节点插值算法, 然后基于节点插值算法将二维九点格式推广至三维.

本文研究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线方程. 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.
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This paper studies the numerical solution of diffusion equation in multi-material radiation hydrodynamics. Diffusion coefficients of multi-material problems are usually discontinuous and anisotropic tensors. In this paper, we will construct an efficient and robust finite volume scheme for diffusion equations with arbitrary anisotropic diffusion coefficients on arbitrary distorted polyhedral meshes. The cell-centered finite volume scheme is widely used in multi-material radiation fluid. In this paper, a class of cell-centered finite volume schemes with second-order accuracy will be derived based on the idea of linearity-preserving, and a new way to construct three-dimensional diffusion scheme will be explored. Firstly, the idea of linearity-preserving is applied to construct the vertex interpolation algorithm on 3D polyhedral meshes, which will be the first second-order accurate vertex interpolation algorithm for 3D large deformation meshes. Then, based on the vertex interpolation algorithm, the 2D nine point scheme is extended to 3D.

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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