基于LU-SGS迭代的DGM隐式方法研究

An Implicit Scheme of Discontinuous Galerkin Method( DGM)Based on LU-SGS Iterative Method

考虑到LU-SGS迭代法已经在基于非结构网格的有限体积法中得到了成功应用,文章借鉴其思想,将其推广到高精度间断Galerkin有限元隐式格式求解中来,并对其性能进行了研究。为了避免隐式算法中对大型稀疏矩阵求逆,采用LU-SGS迭代法,只需要在每步时间推进中沿网格号从前到后和从后到前2次扫描计算即可,并且还能有效降低内存需求。通过对NACA0012翼型和ONERA M6机翼跨声速无粘流动进行数值模拟,计算结果表明:与TVD-RKDG显式时间格式相比,隐式格式所需的迭代步数和CPU时间均得到了很大程度上的减少,并且精度保持不变。

Because the LU-SGS iterative method has been successfully applied to the finite volume method based on unstructured grids, we extend it to the implicit scheme of the high-precision discontinuous Galerkin finite element method and study its properties. To avoid solving the inversion of large sparse matrix in the implicit scheme, we use the LU-SGS iterative method to conduct the forward scan and backward scan computation along the Grid Number at each step of time progression, thus effectively reducing memory requirements. To verify the effectiveness of implicit scheme of the DGM, we simulate the transonic inviscid flow around the NACA0012 airfoil and the ONERA M6 wing and compare the simulation results with the TVD-RKDG computation results. The simulation results, given in Figs. 2, 3 and 5, and their analysis show preliminarily that, compared with the TVD-RKDG computation method, the implicit scheme of the DGM greatly reduces the number of iterations, the CPU time and computational cost, thus increasing the speed of convergence around the flow field and computational efficiency.