一类求解波动方程的加速Schwarz波形松弛方法

An Acceleration Method of Schwarz Waveform Relaxation for Wave Equations

波动方程在声学、电磁学和流体动力学等领域上有着广泛的应用.本文针对波动方程,研究了一类新的Schwarz波形松弛方法.经典Schwarz波形松弛方法是一种迭代方法,在求解波动方程时,特别是当子区域间的重叠量特别小的情形下,迭代次数往往较多,计算量较大.而本文构造的加速Schwarz波形松弛方法,即AitkenSchwarz波形松弛方法与Steffensen Schwarz波形松弛方法,是一种直接方法,它通过构造子区域边界信息的映射矩阵,很大程度地提升了计算性能.文中分别分析了这两种方法的收敛性,并且验证了新方法对于波动方程的可行性.数值算例证实了方法的有效性.

Wave equations are very important in the areas of acoustic, electromagnetic and fluid dynamics. In this paper, we present a new Schwarz waveform relaxation method for wave equations. The classical Schwarz waveform relaxation method, which is an iterative method, costs great, especially when the overlap is very small. While the new Schwarz waveform re- laxation methods constructed in this paper, named Aitken acceleration method and Steffensen acceleration method, are direct methods. The new acceleration methods involves mapping matrices on boundary information of sub-domains, which helps improve the computational ef- ficiency. We first analyze the convergence of the two methods, which indicates the feasibility of the methods for wave equations, and then use numerical experiments to verify the effectiveness of our methods.