In this paper, a space-time generalized finite difference method (ST-GFDM) is proposed to solve the transient Stokes/Parabolic moving interface problem which is a type of fluid-structure interaction problem. The ST-GFDM considers the time dimension as the third space dimension, and the 2D time-dependent Stokes/Parabolic moving interface problem can be seen as a 3D interface problem where the interface is formed by the initial interface shape and the moving trajectory. The GFDM has an advantage in dealing with interface problems with complex interface shape and moving interface in the ST domain. More irregular moving direction, more complex interface shape, and the translation and deformation of interface are analyzed to show the advantage of the ST-GFDM. The interface problem can be transformed into coupled sub-problems and locally dense nodes is used when the subdomain is too small to satisfy the needs of the numbers of the nodes to improve the performance of the ST-GFDM. Five examples are provided to verify the existence of the good performance of the ST-GFDM for Stokes/Parabolic moving interface problems, including those of the simplicity, accuracy, high efficiency and stability.

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