多介质辐射扩散问题的高精度快速求解方法

Numerical simulations for energy transfer in radiation hydrodynamics often need enoumous calculation works, and have difficulties in iteration solving and preseving fundamental mathematical properties. Aimed at overcoming these diffuculties and meeting the needs of practical simulations, studies on the nonlinear discrete schemes and iteration acceleration methods for multi-material radiation diffusion problems on distorted meshes will be carried out in this project, which have both high accuracy and high efficiency. The following contents are included. (1) By taking into account the main features of the radiation diffusion problems, i.e., nonlinearity, strong coupling and discontinuity, some robust cell-centered nonlinear finite volume schemes will be designed, which have high temporal and spatial accuracy, befit irregular meshes and discontinuous diffusion coefficient problems and preserve the fundamental features such as conservation, positivity, coerciveness, and so on, to provide high fidelity simulations for multi-material problem. (2) To deal with the problem of heavy calculations in the complicated multi-dimensional case, some practical adaptive iteration methods matching such nonlinear schemes will be designed, which have superlinear convergent ratio, and preserve the above important features. New program modules will be presented to realize high efficient solutions and acquire remarkable acceleration effects. (3) New inductive reasoning techniques wil be developed, which are different from those used for theoretical analysis for linear schemes, and capable of overcoming the difficulties caused by the nonlinear diffusion operators. Theoretical analysis will be performed on the fundamental properties which include the convergence, stability and unique existence of the solutions of the nonlinear discrete schemes, as well as the convergence accuracy and convergent speed of the iteration methods, to provide computation methods with high confidence to support the building of applied software platforms.

针对辐射流体力学能量传输数值模拟计算量大、迭代求解困难、且难以保持重要数学特性的问题，研究扭曲网格上多介质辐射扩散问题的高效高精度非线性离散格式及迭代加速求解方法。包括：（1）针对辐射扩散问题非线性、强耦合、间断的特点，设计健壮的非线性单元中心型有限体积离散格式，使之具有较高的时间和空间精度，适用于非规则网格和间断扩散系数问题，并保持守恒性、保正性、强制性等重要特性，适用于多介质问题的高保真模拟。（2）针对高维复杂情形计算量大的问题，设计与格式匹配的实用的自适应迭代方法，使之具有超线性收敛速度，并保持上述重要特性；研制新的程序模块，实现问题的高效求解，获得明显的加速效果。（3）发展不同于线性格式理论分析的新归纳论证方法，克服非线性扩散算子带来的难点，给出非线性离散格式解的存在唯一性、收敛性、稳定性和迭代方法的收敛精度、收敛速度等基本性质分析。为应用软件平台建设提供具有高置信度的算法支撑。

辐射流体力学能量传输数值模拟存在精度较低、计算量大、迭代求解困难、且难以保持重要数学特性的问题，本项目面向实际应用需求，开展扭曲网格上多介质辐射扩散问题的高效高精度非线性离散格式及迭代加速求解方法研究。主要成果包括：.（1）针对辐射扩散问题（守恒型非线性扩散问题）非线性、强耦合、间断的特点，利用未知量与非线性离散通量连续，推导出网格边界处未知量的表达式，获得扩散通量的高精度离散，设计了健壮的非线性单元中心型有限体积离散格式，格式具有较高的时间和空间精度，适用于非规则网格和间断扩散系数问题，并保持局部守恒性、保正性、强制性等重要特性，适用于多介质问题的高保真模拟。.（2）针对高维复杂情形计算量大的问题，利用“线性化-离散”方法，发展了新的网格边界未知量线性化技术，设计了与非线性离散格式匹配的实用的自适应迭代方法—自适应Picard-Newton（PN）和无导数PN迭代加速方法。方法具有超线性（二次）收敛速度，并保持上述重要特性；研制了新的程序模块，实现了问题的高效求解，获得了明显的加速效果。.（3）发展了适用于扩散系数连续和间断情形的非线性有限体积格式的新的归纳论证技术，克服非线性扩散算子带来的难点，从理论上严格证明了非线性有限体积格式和迭代方法的基本性质，包括格式解的存在性、唯一性、收敛性和稳定性以及PN迭代方法的保正性、收敛速度和精度。.数值实验验证了新方法的高精度、高效率和健壮性。.本项研究在多介质辐射扩散问题新的非线性有限体积格式和迭代方法设计和理论分析、新的程序模块研制和数值实验方面，取得了确定的进展，提高了数值模拟的精度和效率。为提高多维辐射流体力学整体模拟能力提供了具有高置信度的高性能多介质辐射扩散计算方法。.