The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.
翻译:辐射传输方程式是运输理论和应用中的一个基本方程式,在固定的单速度情况下是一个五维的PDE,在数字模拟中造成了很大的困难。为了解决这一瓶颈问题,我们首先使用离散坐标技术将散射术语分解,这是一个对角变量的有机体,形成半分解双曲系统。然后,我们通过不连续的加列尔金(DG)方法与稀疏的网格方法结合,使空间分解。最后的线性系统由块高斯-潮汐迭代法解开。计算复杂性和误差分析是详细开发的,这表明新方法比原始离散坐标DG方法更有效。进行了一系列数字结果,以验证拟议方法的趋同行为和有效性。