Tsunami Generation due to Supershear Earthquakes: A Case Study

<p>The 28 September 2018 Mw 7.5 Sulawesi strike-slip earthquake generated an unexpected tsunami with devastating consequences. Since such strike-slip earthquakes are not expected to generate large tsunamis, the latter’s origin remains much debated. A key notable feature of this earthquake is that it ruptured at supershear speed, i.e., with a rupture speed greater than the shear wave speed of the host medium. Dunham and Bhat (2008) showed that such supershear ruptures, in half-space, produce two shock fronts (or Mach fronts) corresponding to an exceedance of shear and Rayleigh wave speeds. The Rayleigh Mach front carries significant vertical velocity along its front. We couple the ground motion produced by such a supershear earthquake to a 1D non-linear shallow water wave equation that accounts for both the time-dependent bathymetric displacement as well its velocity. We use an extension of Fourier-based PDE solvers called the Fourier Continuation (FC) method to numerically solve the system. The FC method enables high-order convergence of Fourier series approximations of non-periodic functions by resolving the well-known Gibbs “ringing” effect.  FC-based solvers offer limited numerical dispersion, high-order accuracy and mild CFL conditions—making them ideal to solve this system. Using the local bathymetric profile of Palu bay around the Pantoloan harbor tidal gauge, we have been able to clearly reproduce the observed tsunami with minimal tuning of parameters. We conclude that the Rayleigh Mach front, generated by a supershear earthquake combined with the Palu bay geometry, caused the tsunami.</p>

Two-dimensional local Fourier image reconstruction via domain decomposition Fourier continuation method

The MRI image is obtained in the spatial domain from the given Fourier coefficients in the frequency domain. It is costly to obtain the high resolution image because it requires higher frequency Fourier data while the lower frequency Fourier data is less costly and effective if the image is smooth. However, the Gibbs ringing, if existent, prevails with the lower frequency Fourier data. We propose an efficient and accurate local reconstruction method with the lower frequency Fourier data that yields sharp image profile near the local edge. The proposed method utilizes only the small number of image data in the local area. Thus the method is efficient. Furthermore the method is accurate because it minimizes the global effects on the reconstruction near the weak edges shown in many other global methods for which all the image data is used for the reconstruction. To utilize the Fourier method locally based on the local non-periodic data, the proposed method is based on the Fourier continuation method. This work is an extension of our previous 1D Fourier domain decomposition method to 2D Fourier data. The proposed method first divides the MRI image in the spatial domain into many subdomains and applies the Fourier continuation method for the smooth periodic extension of the subdomain of interest. Then the proposed method reconstructs the local image based on L2 minimization regularized by the L1 norm of edge sparsity to sharpen the image near edges. Our numerical results suggest that the proposed method should be utilized in dimension-by-dimension manner instead of in a global manner for both the quality of the reconstruction and computational efficiency. The numerical results show that the proposed method is effective when the local reconstruction is sought and that the solution is free of Gibbs oscillations.

A Fourier Continuation Method for the Solution of Elliptic Eigenvalue Problems in General Domains

We present a new computational method for the solution of elliptic eigenvalue problems with variable coefficients in general two-dimensional domains. The proposed approach is based on use of the novel Fourier continuation method (which enables fast and highly accurate Fourier approximation of nonperiodic functions in equispaced grids without the limitations arising from the Gibbs phenomenon) in conjunction with an overlapping patch domain decomposition strategy and Arnoldi iteration. A variety of examples demonstrate the versatility, accuracy, and generality of the proposed methodology.