高阶DG格式多重网格方法构造中的不同隐式策略影响研究

DG方法是一种非常具有潜力的高精度方法,但其在对复杂外形的数值模拟方面仍存在内存需求量大、计算量巨大等不足.为了进一步提高DG方法求解Euler方程的效率,在传统p型多重网格的基础上,结合LU-SGS和GMRES两种隐式迭代方法,研究其整体加速性能.p型多重网格方法通过对不同阶次多项式近似解进行递归迭代求解,来达到加速收敛的目的.高阶近似(p＞0)使用显式龙格库塔格式,最低阶近似(p=0)使用隐式格式.对NACA0012翼型和ONERA M6机翼跨音速无粘流动进行数值模拟,结果表明:与显式TVD-RKDG时间格式相比,DG(p0)层上采用LU-SGS和GMRES的p型多重网格方法收敛速度均得到明显提高,且GMRES迭代法性能最佳,LU-SGS迭代法次之.

Study of Different Implicit Schemes for High Order DG Multigrid Method

Considering LU-SGS and GMRES have been successfully applied into unstructured grid finite volume method as two main implicit iterative methods, they are combined with p-multigrid method in this paper and the overall acceleration performance is studied. The convergence acceleration of p-multigrid method is achieved through the recursive iterative solving of different order polynomial approximations. In order to achieve better convergence effect, implicit scheme is implemented on the lowest-order approximation while explicit schemes are implemented on the higher-order approximations. The transonic inviscid flow around NACA0012 airfoil and ONERA M6 wing were simulated. The numerical results show that, compared with the explicit TVD-RKDG method, convergence rates for the p-type multigrid methods, which adopt LU-SGS or GMRES scheme on DG (P₀) layer, have been significantly improved, while the performance of GMRES iterative method is best and LU-SGS iteration is secondary.