解Helmholtz方程的共轭梯度法三角阵预处理器

利用对应微分运算的差分运算矩阵,提出了差分求解Helmholtz方程的三角阵预处理器,阐述了在算子离散的过程中而非离散后构造高效率预处理器的基本思想.利用二阶频域Mur吸收边界条件下的二维导体柱散射的模型问题,结合法方程最小余量预处理共轭梯度法(PCGNR)验证了该预处理器的有效性.数值结果表明了该预处理器能够在网格精度提高和吸收边界趋远时,相对常规共轭梯度法具有降低计算复杂度的效果,而存储复杂度并没有提高.同时,也揭示了二维散射问题在网格精度与吸收边界距离一定的情况下,用不同共轭梯度法求解时,由于迭代次数变化较少,计算量几乎随未知量线性增长.

A Triangular Matrix Preconditioner of the Conjugate Gradient Method for Solving Helmholtz Equation

Based on the difference matrices corresponding to differential operators,a triangular matrix preconditioner is presented in this paper.A basic idea to construct high efficient preconditioners in the process of operator discretization,not after the process,is emphasized.The validity of the preconditioner is verified by means of preconditioned conjugate gradient on normal equation with minimization of the residual(PCGNR) and the model problem of two dimensional scattering from a conducting cylinder with the second order Mur's absorbing boundary condition.The numerical results show that,the computational complexity of the preconditioned method,by contrast with that of conventional conjugate gradient method,can be decreased for finer mesh sizes and farther absorbing boundary,with the same storage complexity.It is also revealed from the results that the arithmetic operations almost linearly grow with the unknowns under the condition of fixed mesh size and absorbing boundary distance,due to the less iteration number changing of different versions of the conjugate gradient methods.