Akademik Veri Yönetim Sistemi
Yükseköğretim Kurumları Destekli Proje, BAP Araştırma Projesi, 2025 - 2027
The runup of tidal waves or tsunamis generated by impulsive geophysical events on a beach are known to cause extensive inundation and loss of life. In spite of their destructive powers, tsunamis are not well understood. In fact, on the tsunami prone shores, the preventive measures, such as building high-walls or installing warning systems, are still based on historical data. Faced with such catastropic events, the prediction of runup heights and understanding the mechanisms of inundation have been an important research area. In this research effort, numerical as well as experimental studies play important roles, the former providing additional freedom in testing the effects of various parameters. The nonlinear inviscid shallow-water wave equation has been widely used in modeling the propagation of long water waves such as an incoming tsunami. In addition, solitary wave profiles are used both experimentally and numerically as a model for far field tsunamis in literature due to their simple description based on one parameter. Their evolution and runup under the shallow-water equation are studied numerically to provide insight into the mechanisms of tsunami inundation. The factors affecting the success of the numerical approaches to tsunami modeling can be collected in three main headings: These are (i) the numerical scheme for approximating derivatives and the treatments of the (ii) moving shoreline boundary and (iii) the open sea boundary. In the numerical approaches of wide-spread use, the finite difference method plays the central role in approximating the derivatives. The open sea boundary treatment by using an absorbing (transparent) boundary condition based on the characteristic paths and speeds of the wave propagation is generally accepted as satisfactory. Even though the moving boundary where the sea surface intersects the shoreline and the water depth vanishes, is discarded and the dynamics of the shoreline motion is ignored in the early work, later work have addressed this issue of moving boundary by employing various techniques. In this study, the reduced wave equations will be solved numerically in time using the Discontinuous Galerkin Spectral Element method as a numerical approach to spatial partial derivatives. Unlike the finite difference method, the Spectral Element method uses the model equations in their variational (weak) form by expanding the model unknowns and partial spatial derivatives in terms of spatially directed prime functions within each sub-element of the solution geometry. The Discontinuous Galerkin method handles the interactions between sub-element geometry using numerical flux relations, which are required without using the continuity constraint. In this study, Legendre-Gauss nodal-based polynomial expansions will be used. This allows the spatial integrals constituting the variational form to be calculated with high-order accuracy using the numerical method of Gaussian quadrature. The moving boundary formed by the wave run-up tip will be addressed comparatively using two methods: (1) a mathematical transformation based on front fixing and (2) a stepwise total linearization using small deviations of the geometric parameters and flow variables. The offshore boundary will be rendered transparent to waves originating in the solution region using wave characteristic directions and speeds, as is commonly used in the literature.