时间谱方法中的高效GMRES算法

Efficient GMRES algorithm in time spectral method

研究了时间谱方法求解周期性非定常流场的计算效率,并对时间谱方法应用于周期性非定常流动的隐式求解方法进行探讨。当采样点数增加或减缩频率增大时,时间谱方法对应的雅可比矩阵对角占优性质迅速恶化,导致很多传统的迭代方法失效。为了解决上述问题,论文采用带预处理的广义极小残差(GMRES)算法来提高雅可比系数矩阵的计算收敛性。使用时间谱方法对NACA0012翼型强迫振荡算例进行计算,并与时域差分方法的计算效率和精度进行对比。研究表明在保证计算精度的同时,时间谱方法普遍可将计算效率提高一个量级左右。对于跨声速周期性流动,广义极小残差算法不论是稳定性还是收敛性都优于对称SGS迭代算法。

In this paper,the computational efficiency of the time-spectral method for solving the periodic unsteady flow field is studied,and the implicit method of time spectral method for solving the periodic unsteady flow is discussed.When the number of sampling points increases or the reduced frequency magnifies,the diagonal dominant property of the Jacobian matrix corresponding to the time spectral method deteriorates rapidly,resulting in the failure of many traditional iterative methods.In order to solve the problems above,the generalized minimum residual（GMRES）algorithm with preprocessing is used to improve the computational convergence of the Jacobian matrix.The time spectral method is used to compute the NACA0012 airfoil forced oscillation,and the computational efficiency and accuracy is compared with that of the time-domain difference method.The results show that the time spectral method can generally improve the computational efficiency an order of magnitude with saturated computational accuracy.For the transonic periodic flow,the GMRES algorithm is superior to SGS iterative algorithm both in stability and computational convergence.