含边界层与界面层的输运方程数值算法研究

Transport equations are widely used in a lot of physical and biological problems. Due to the dependence of multivariables and high dimensionality, its numerical simulations are difficult because of the large computational cost after discretizations. Therefore people approximate the solution by its diffusion limit when the mean free path is small. In real applications, the mean free path can vary widely in different physical domains. When the transport and diffusion regions connect, or there are reflections and refractions at the interfaces, there may exist boundary/interface layers in the solution. Existing schemes require enough meshes in each interface layers to capture the fast changes in the solutions. Designing numerical schemes that can capture the boundary and interface layers by coarse meshes, so that to reduce many of computational costs, is one of the hot topic in this area. This project studies the velocity and space discretizations for the neutron transport equation with discontinuous coefficients. On the one hand, we give the proper discrete reflection and refraction operators, which make the discrete ordinate equation and the original integral equation have the same interface conditions for their diffusion limit; on the other hand, we design new space discretizations, which can capture the high dimensional boundary and interface layers of the neutron transport equation with coarse meshes. Furthermore, we will extend the methods that can capture the boundary and interface layers to more general collision kernels and two group transport equations.

输运方程在物理、生物等众多领域中都有广泛应用。由于方程解依赖于多个变量，维数较高，这给数值求解带来很大困难，计算量太大。因此人们在平均自由程小时往往采用扩散极限近似。在实际应用中，不同物理区域粒子的平均自由程可能差别很大。当输运区域和扩散区域连接，或者是界面处存在反射、折射时，解中可能含边界层与界面层。现有的算法需要在每个界面层内放足够多的网格点来捕捉解的快速变化。设计能用粗网格捕捉边界层、界面层的算法以大大减少计算量是该领域的研究热点之一。 本项目研究含有间断系数的中子输运方程的速度离散和空间离散，一方面给出合适的界面处的离散反射折射算子，使得速度离散方程具有和积分方程一致的扩散极限界面连接条件；另一方面设计新的空间离散格式，使得算法可以用粗网格捕捉到中子输运方程解中的高维边界层、界面层。并进一步把捕捉高维边界层、界面层的算法扩展到更一般的的碰撞核和双群输运方程。

输运方程在物理、生物等众多领域中都有广泛应用。由于方程的解依赖于6个变量，维数较高，计算量太大给数值求解带来很大困难。因此人们使用一些近似模型进行求解，比如在平均自由程小时采用扩散方程近似。在实际应用中，由于不同物理区域粒子的平均自由程可能差别很大。当输运区域和扩散区域连接、界面处存在反射折射时，不同区域连接的位置可能出现边界层和界面层。现有的算法需要在每个界面层内放足够多的网格点来捕捉解得快速变化，从而更进一步增加了计算难度。设计能用粗网格捕捉边界层、界面层的算法可以大大减少计算量，是该领域的研究热点之一。.本项目研究含有间断系数的中子输运方程的速度离散和空间离散。项目对各项同性散射的中子输运方程提出了不仅在计算区域内部，而且在边界层、界面层处具有渐进保持性质的算法，大大提高了计算效率。同时对各向异性散射也给出了散射核合理的速度离散，使得扩散极限的扩散系数不受速度离散方式的影响。所得算法可以抓住高维边界层界面层，有很强的扩展应用价值，可以从核反应堆中子输运扩展到X光输运，对后继应用研究起到奠基作用。