高维扩散方程的非线性保正有限体积格式设计及其理论分析

For the demand of numerical simulation of Z-pinch driven fusion-fission hybrid reactor, this project will study several accurate and positivity-preserving nonlinear finite volume schemes for high dimensional diffusion equation on unstructured meshes with strong distortion under the extreme physical conditions. The main research contents include the following: (1) construction of the accurate and positivity-preserving finite volume scheme for two dimensional diffusion equation on unstructured meshes with strong distortion; (2) construction of the accurate and positivity-preserving finite volume scheme for diffusion equation on the unstructured polyhedral meshes; (3) design of an efficient solver for the resulting nonlinear system; (4) theoretical analysis of the proposed positivity-preserving finite volume schemes. The research will provide the accurate, efficient and robust algorithm for solving the energy equation and magnetic diffusion equation in the numerical simulation of magnetohydrodynamics, and improve the ability of numerical simulation of Z-pinch driven fusion-fission hybrid reactor design in our country. In addition, the results of this project can be applied to some important areas such as hydrological geology and reservoir simulation, and is of great theoretical significance and application value.

本项目以Z箍缩驱动聚变-裂变混合能源堆设计研究中的数值模拟需求为牵引，在极端物理条件下，研究非结构强扭曲网格上高维扩散方程的高精度非线性保正有限体积格式，主要包括 ：（1）强扭曲网格上二维扩散方程的高精度保正格式构造；（2）非结构多面体网格上扩散方程的高精度保正格式研究；（3）非线性代数系统的高效解法器设计；（4）扩散方程非线性保正格式的理论分析。本项目的研究成果可以为大规模辐射磁流体力学科学计算程序研制过程中能量方程和磁场演化方程的求解提供高精度、高效、保物理且鲁棒的算法，从而为提高我国在Z箍缩驱动混合能源堆设计方面的数值模拟能力作贡献。此外，它还可以应用于水文地质学和油藏模拟领域等一些关系国家民生的重要领域中，具有重大的理论研究意义和应用价值。

本项目以Z箍缩驱动聚变-裂变混合能源堆设计研究中的数值模拟需求为牵引，在极端物理条件下，系统研究了二三维非结构网格上非线性辐射扩散方程的高精度有限体积格式，主要包括：（1）单元中心型保正格式和节点型保正格式；（2）非线性保极值格式；（3）保正解耦型离散对偶有限体积格式；（4）非线性辐射扩散方程高效迭代加速算法；（5）改进型Q1有限体积元格式及其强制性分析。本项目的研究成果为大规模辐射磁流体力学科学计算程序研制过程中能量方程和磁场演化方程的求解提供了高精度、高效、保物理且鲁棒的算法，为提高我国在Z箍缩驱动混合能源堆设计方面的数值模拟能力提供了支撑。