RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES6004, doi:10.2205/2006ES000216, 2006
[72] The objective of calculations, the results of which are described below, was the study of different approaches to provide the accuracy of algorithms. With this goal in mind the calculations were performed using the methods of different order of approximation and grids with different resolution. It was already shown in the first half of this paper that dispersion effects in the problem of the propagation of a solitary wave in the "wash-tub" basin appear insignificant. Thus, the further consideration is based mainly on the material obtained using classical equations of the shallow water theory.
[73] The calculations were carried out using the schemes of MacCormac and Adams and a sequence of grids with sizes decreasing in the OY direction (the number of points was equal to Ny=129 (Dx = 2h0), Ny=257 (Dx = h0), Ny=513 (Dx = h0/2), Ny=1025 (Dx = h0/4) ). The first scheme has the second order of approximation at internal points of the area. The condition of reflection at the wall (at the first node) was approximated by the first order. As to the Adams scheme, the authors used two versions. In the first of them at the second node and at ( N-1 ) node located near the boundary over the OY -axis, the first spatial derivatives in nonlinear terms of equations were approximated using the second order (central difference calculated over three points), while at the other internal nodes, the equations were approximated using the fourth order based at five points. In the second version the fourth order of approximation at near boundary nodes (using a five-point template displaced to the interior of the area) was conserved. The boundary condition at the first node in both cases was approximated using the second order.
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[75] A solitary wave with an amplitude of 1 m, which at the initial time moment was located over gauge 8, splits into two. One of them ("right") propagates to increasing depths and after passing gauges 9 and 10 leaves the basin. The second ("left") propagates to the coast. The "right" wave almost immediately interacts with a sharp break line of the topography, which leads to a distortion of the trailing front of the leading wave, which is manifested at gauges 9 and 10. In the final fragments of these records one can see a flat wave generated by the wave transformation over the shelf area (depth is 1000 m) reflected from a flat coastal slope and from a steep break line of topography at the end of the shelf area. Their interaction is clearly seen in the record of gauge 6, while the record at the next point (gauge 7) shows the separation of the wave mentioned above, which propagates in the seaward direction.
[76] Coastal gauges demonstrate the interaction of the incident wave with the coastal wall. During this interaction, the wave height increases approximately by a factor of 3 (gauge 1). The coastal gauges also demonstrate the formation of the wave reflected from the coast (gauges 2 and 4), propagation of the wave in the direction of increasing depths, and its corresponding transformation during multiple interactions with other components of the wave field and break lines of bottom topography (gauges 5 and 7).
[77] Deep water gauges do not record any difference in the characteristics of the wave field caused by different computational algorithms, thus, an increased order of the Adams scheme does not result in any advantages. No differences are observed in the results obtained using the Adams scheme within the nonlinear dispersion models and classical model of shallow water. This fact makes possible considering in the further description the latter ones as reference models. However, we note that the corresponding algorithm requires large computational resources and more sophisticated realization than the algorithm of the MacCormac scheme.
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Citation: 2006), Principles of numerical modeling applied to the tsunami problem, Russ. J. Earth Sci., 8, ES6004, doi:10.2205/2006ES000216.
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