We propose novel numerical methods for solving a class of high-order hyperbolic PDEs on general geometries, which involve 2nd-order derivatives in time and up-to 4th-order derivatives in space. These PDEs are widely used in modeling thin-walled elastic structures such as beams, plates and shells, etc. High-order spatial derivatives together with general geometries bring a number of challenges for any numerical methods. In this work, we resolve these challenges by discretizing the spatial derivatives in domains with general geometries using the composite overlapping grid method. The discretization on overlapping grids requires numerical interpolations to couple solutions on different component grids. However, the presence of interpolation equations causes weak numerical instability. To address this, a fourth-order hyper-dissipation term is included for stabilization. Investigation of incorporating the hyper-dissipation term into several time-stepping schemes for solving the semi-discrete system leads to the development of a series of algorithms. Carefully designed numerical experiments and two benchmark problems concerning the Kirchhoff-Love plate model are presented to demonstrate the accuracy and efficiency of our approaches. This work shows that finite difference methods on overlapping grids are well-suited for solving high-order PDEs in complex domains for realistic applications.

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