非规则网格上各向异性扩散问题的高性能计算方法

Diffusion problems in multi-dimensional radiation magneto hydrodynamics are often characterized with nonlinearity, discontinuity, strong coupling and high anisotropy, and often need to be solved on severely deformed meshes. Compared with isotropic problems, there exist more difficulties on keeping perfect accuracy and embodying physical properties in the numerical solution of anisotropic diffusion problems. Aimed at meeting the needs of practical simulations, studies on the high performance computation methods for anisotropic diffusion problems with such characters on irregular meshes will be carried out in this project. The following contents are included. (1) For the purpose of being adaptive to multi-physical coupled procedures, new robust cell-centered finite volume schemes will be designed for the anisotropic diffusion problems on arbitrary polygonal meshes, which will have high accuracy and preserve the fundamental physical features such as conservation and so on. (2) Nonlinear schemes for nonlinear anisotropic diffusion problems will be studied, and robust iteration methods matching such nonlinear schemes and having superlinear convergent ratio will be designed to realize the efficient solution of the nonlinear problems. (3) Theoretical analysis will be made on the stability and convergence properties of the discrete schemes and the convergent ratio and accuracy properties of the iteration schemes, also program modules will be developed, which will provide both diffusion computation methods with high efficiency and high accuracy and their theoretical supports for the related numerical simulations in the practical multi-dimensional radiation magneto hydrodynamics applications. The main innovation and advantage of the project is that both the anisotropy of the diffusion and the irregularity of the meshes will be taken into account, highly efficient computation methods will be designed, and high performance numerical simulations will be accomplished.

多维辐射磁流体力学问题中的扩散，通常具有非线性、间断、强耦合和强各向异性的特点，并需在大变形网格上求解；各向异性问题的求解与各向同性相比，在保持精度和物理性质等方面存在更大困难。本项目针对应用问题需求，开展具有上述特征的非规则网格上各向异性扩散问题的高性能计算方法研究。(1)从适应多物理耦合的角度出发，基于任意多边形网格，针对各向异性扩散问题，设计新的健壮的单元中心型有限体积格式，使之具有较高精度和保持守恒性等重要物理性质；(2)研究非线性扩散的非线性格式，设计与之匹配的具有超线性收敛速度的健壮的迭代方法，实现问题的高效求解；(3)进行离散格式的稳定性、收敛性和迭代法的收敛速度、精度等理论分析，研制程序模块，为多维辐射磁流体力学应用中的相关数值模拟提供高效高精度的扩散计算方法及其理论支持。项目的主要特色和创新在于：兼顾扩散的各向异性和网格的非规则性，设计高效计算方法，实现高性能数值模拟。

多维辐射磁流体力学问题中的扩散，通常具有非线性、间断、强耦合和强各向异性的特点，并需在大变形网格上求解；各向异性问题的求解与各向同性相比，在保持精度和物理性质等方面存在更大困难。本项目针对应用问题需求，开展具有上述特征的非规则网格上各向异性扩散问题的高性能计算方法研究。主要成果包括：. (1) 从适应多物理耦合角度出发，兼顾网格非规则性和扩散各向异性的特点，利用变量向量的合理方向分解给出扩散通量的高精度离散，给出基于通量连续的新节点值计算方法，设计了各向异性扩散问题新的健壮的单元中心型有限体积格式，使之适应于任意多边形网格和处理间断系数问题，具有较高精度和保持守恒性等重要物理性质。对间断系数问题，在含随机网格在内的多种扭曲网格上获得高于一阶的收敛性。. (2) 研究非线性扩散的非线性离散格式。构造了各向异性扩散具有上述良好性质和较高时空精度的非线性有限体积格式；基于高度非线性的两温限流模型，研究非平衡辐射扩散问题的渐近保持格式。设计与格式匹配的具有超线性收敛速度的健壮的迭代方法，实现了非线性问题的高性能求解。. (3) 开展离散和迭代格式的性质分析。发展了适用于一般网格的广义离散Poincarè不等式，利用二次型表达式，证明各向异性扩散格式的稳定性和收敛性。发展了不同于线性格式理论分析的新归纳论证方法，克服守恒型非线性扩散算子带来的困难，证明了非线性离散格式解的稳定性、收敛性和迭代方法的收敛速度和精度。新方法可推广应用于若干非线性问题非线性格式的理论分析。. (4) 设计了三维多面体网格上各向异性扩散问题的保正和保离散极值原理格式，格式适用于扭曲网格（含表面为非平面的网格）和处理间断系数问题。研究了任意多边形网格上输运方程的单元中心型有限体积格式，提高了输运各向异性问题的计算效率。. 研制了新的程序模块，开展数值实验，为多维辐射磁流体力学应用中的相关数值模拟提供了高效高精度的扩散计算方法及其理论支持。.