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Phase reduction theory has been applied to many systems with limit cycles;however, it has limited applications in incompressible fluid systems. This isbecause the calculation of the phase sensitivity function, one of thefundamental functions in phase reduction theory, has a high computational costfor systems with a large degree of freedom. Furthermore, incompressible fluidsystems have an implicit expression of the Jacobian. To address these issues,we propose a new algorithm to numerically calculate the phase sensitivityfunction. This algorithm does not require the explicit form of the Jacobianalong the limit cycle, and the computational time is significantly reduced,compared with known methods. Along with the description of the method andcharacteristics, two applications of the method are demonstrated. Oneapplication is the traveling pulse in the FitzHugh Nagumo equation in aperiodic domain and the other is the K\'arm\'an's vortex street. The responseto the perturbation added to the K\'arm\'an's vortex street is discussed interms of both phase reduction theory and fluid mechanics.

Jacobian-free algorithm to calculate the phase sensitivity function in the phase reduction theory and its applications to K\'arm\'an's vortex street