[S-CG52] Dynamics in mobile belts

[SCG52-P01] Linear-time arithmetic for wave-equational integral equation methods

*Daisuke Sato¹ (1.Disaster Prevention Research Institute, Kyoto University)

Keywords:Simulations, Dynamic problems, Integral equation methods

The integral equation method is a generic term referring to the simulation methods that use the fundamental solution (Green's function) of the given partial differential equational problem. It is popular in geophysics, as typified by the inversion methods using the Okada model, while its problem is the costly simulation time and memory storage requirement. The large scale computation is demanded (e.g. Ozawa et al. 2021), and the acceleration algorithm is also (e.g. Ohtani et al., 2011, Ando, 2016). The static problem implementation considers the spatial convolution in the method and has the associated acceleration algorithm, such as the fast multipole method (Rokhlin, 1985), fast comparably to the finite difference/element methods; that is, the associated computation time is scaled by the number of elements (the linear time algorithm). However, it is not comparably successful for the coseismic (elastodynamic) simulation which involves the spatiotemporal convolution, largely because of the distributed singular points in the associated wave-equational Green's function (Born et al., 2003). Lapusta et al. (2000) provide a spectral-based algorithm achieving the log-linear order for the planar faults, which are recently extended to involve the small roughness on the fault (Romanet and Ozawa, 2021).

A versatile fast algorithm is still desired for the dynamic integral equation methods to overcome the limitation of the simulated fault geometry, and Sato and Ando (2021) have developed the fast domain partitioning hierarchical matrices (FDP=H-matrices) applicable to arbitrary fault geometry with log-linear order time and memory usage, along the line of the fast domain partitioning method (Ando et al., 2008; Ando, 2016) and hierarchical matrices (Hackbusch, 1999). We further extend FDP=H-matrices, and now construct a linear time algorithm, the FDP plane-wave time-domain (FDP-PWTD) method. The new finding of this study is extended M2M, L2L, M2L formulas for the time-domain wave-equational problems, which are originally obtained in the FMM only for the static problems. Combining them conceivable from the plane-wave time-domain method (Ergin et al. 1999) with the algorithm of FDP=H-matrices, we can develop the FDP-PWTD method. In the presentation, we discuss the algorithm architecture and benchmark using the two-dimensional planar element configuration, where the plane-wave expansion becomes error-free. We may discuss its application to two-dimensional anti-plane problems and a simple three-dimensional wave equation case, the implication of which is an intriguing application to many-particle problems.