Arnoldi Method and Its Application in Flow Stability Problem
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摘要: 流动稳定性问题常常归结于巨型非对称矩阵特征值问题。多数求解巨型非对称矩阵特征问题的算法均是经基本的Arnoldi算法演化而来。首先简述基本的Arnoldi算法; 其次简述基于Arnoldi算法的几类变体, 如显式重启Arnoldi算法，隐式重启Arnoldi算法与多重隐式重启Arnoldi算法; 最后基于Arnoldi算法及其变体结合谱位移技术求解计算流动稳定性问题, 并通过数值实验比较可知结合谱位移技术的多重隐式重启Arnoldi算法的求解效率最高。
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关键词:
- Arnoldi算法 /
- 特征值 /
- Hessenberg矩阵 /
- 流动稳定性
Abstract: The problem of flow stability is often attributed to the eigenvalue problem of giant asymmetric matrix. Most algorithms for solving the eigenvalue problems of giant asymmetric matrices are evolved from the basic Arnoldi algorithm. Firstly, the basic Arnoldi algorithm was briefly described. Secondly, several variants based on Arnoldi algorithm were briefly described, such as explicitly restarted Arnoldi algorithm, implicitly restarted Arnoldi algorithm and multiple implicitly restarted Arnoldi algorithm. Finally, the Arnoldi algorithm and its variants, combined with spectral displacement technology to solve the problem of flow stability. Through numerical experiments, it is found that the multiple implicitly restarted Arnoldi algorithm combined with spectral displacement technology has the highest efficiency.-
Key words:
- Arnoldi algorithm /
- eigenvalue /
- Hessenberg matrix /
- flow stability
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图 3 多重隐式重启Arnoldi算法流程图
Figure 3. Flow chart of multiple implicitly restarted Arnoldi algorithm
图 4 给定谱位移参数σ=0.3情形下3种不同的Arnoldi方法求解的Poiseuille流扰动特征值问题的特征谱
Figure 4. Characteristic spectra of three different Arnoldi methods for solving the perturbation eigenvalue problem of poiseuille flow at the spectral displacement parameter σ=0.3
图 5 3种不同的Arnoldi方法求解的Poiseuill流扰动特征值问题所对应的残差
Figure 5. Residual corresponding to the perturbation eigenvalue problem of poiseuille flow solved by three different Arnoldi methods
图 6 不同谱位移参数σ=0.1~0.9情形下多重隐式重启Arnoldi方法得到的特征谱
Figure 6. Characteristic spectra obtained by multiple implicitly restarted Arnoldi method under different spectral displacement parameters σ=0.1~0.9
图 7 3种方法分别求解Ma=2.5高超声速边界层稳定性与函数eig的比较
Figure 7. Comparison of three methods for solving hypersonic boundary layer stability at Ma=2.5 with function eig
图 8 Ma=6高超声速6°攻角圆锥边界层背风流向涡不稳定模态特征函数分布
Figure 8. Characteristic function distribution of unstable modes of vortices in the leeward side of conical boundary layer at 6° angle of attack and hypersonic speed Ma=6
表 1 3种方法求解不可压缩二维流动比较
Table 1. Comparison of three methods for solving two-dimensional incompressible flow
methods CPU/s ITE m restarted Arnoldi * * 25 implicitly restarted Arnoldi 0.445 065 35 25 multiple implicitly restarted Arnoldi 0.218 831 1 29 
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表 2 3种方法分别求解来流Ma=2.5的平板边界层流动比较
Table 2. Comparison of three methods for solving the boundary layer on a flat plate at incoming Mach number Ma=2.5
methods CPU ITE m restarted Arnoldi 25.386 330 s 56 50 implicitly restarted Arnoldi 11.892 779 s 5 50 multiple implicitly restarted Arnoldi 11.434 565 s 1 52
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表 3 3种方法求解不可压缩二维流动比较
Table 3. Comparison of three methods for solving two-dimensional incompressible flow
methods CPU/s ITE m E restarted Arnoldi 186.418 92 2 50 0.050 060 74-0.003 5251i implicitly restarted Arnoldi 513.948 11 1 50 0.048 060 73-0.003 466 2i multiple implicitly restarted Arnoldi 137.648 17 1 55 0.049 090 03-0.003 381 3i
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