辐射输运方程保物理特性的多尺度HOLO算法研究

典型应用问题中辐射输运方程的数值模拟极具挑战性，尤其是计算效率与国际同行仍有量级差距。一方面辐射与物质强耦合源项导致经典的源迭代求解缓慢甚至失败；另一方面空间离散格式要求能同时满足高精度与保物理特性（渐近保持性质、保正性、守恒性），而现有的格式数值解出负，导致长时间下守恒误差大，且时间步长缩小，进一步降低计算效率。. 本项目针对上述关键难点，开展拉氏框架下具有保物理特性的多尺度高阶低阶(HOLO)算法研究：基于HOLO算法获得精度和效率的平衡；通过保物理特性的离散格式，确保大时间步长下的高精度求解；通过低阶方程的快速求解进一步提高整体求解效率。主要研究内容包括：构造与输运方程相容的低阶扩散校正方程，实现HOLO系统的非线性耦合求解；发展HOLO系统具有保物理特性的高精度离散格式；发展低阶扩散方程的快速求解方法。最后，通过典型模型验证算法的有效性，为实际的应用研究提供算法和基础。

The numerical simulation of radiation transport equation in typical application problems is very challenging. Especially the computational efficiency still lags behind the international counterparts. On the one hand, for the strong coupling source term of radiation and matter, the classical source iteration is very slow or even fail. On the other hand, the spatial discretization scheme is required to satisfy both high accuracy and physical properties (asymptotic preserving, positive preserving and conservation properties). However the existing schemes can produce negative numerical solutions. As a result, it leads to a large conservation error after a long time calculation, and the time step is reduced, which further reduces the computational efficiency.. Aiming at the above key difficulties, this project studies a multi-scale high-order and low-order (HOLO) algorithm with preserving physical properties in Lagrangian frame. The balance between the accuracy of HO equations and the efficiency of LO equations is obtained based on HOLO algorithm. The discrete scheme with preserving physical properties is used to ensure the high-precision solution in large time step. Through the fast solution of LO equations, the overall efficiency is further improved..The main research contents include: constructing LO diffusion correction equations consistent with HO transport equations and achieving the nonlinear coupling solution of the HOLO system; the development of high-precision discrete schemes with physical properties based on HOLO system; developing fast methods for solving LO diffusion equations. Finally, the effect of the algorithm is verified by typical model examples, which provides the basis for further application research.