RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES6004, doi:10.2205/2006ES000216, 2006

5. Analysis of the Accuracy of Algorithms on the Example of the Problem of Tsunami Waves Transformation

[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.

2006ES000216-fig11
Figure 11
[74]  First of all, we shall describe the general characteristics of this quasi-one-dimensional process, which are shown in Figure 11 as gauge records calculated along the pathway of wave propagation to the coast (gauges 1-7) and in the seaward direction (gauges 8-10).

[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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Figure 12
[78]  The effects of different approaches to the control of accuracy of calculations of waves at the coastal point are presented in Figure 12. It is clearly seen that an increase in the accuracy of the computational algorithm allows us to obtain acceptable results without decreasing the size of the grid (thin solid line in fragment (b)). This effect is observed both for the incident and reflected waves. The MacCormac scheme (fragment (a)) responses stronger to roughening of the grid. It is easy to see that the calculations performed using these grids can be used only for approximate estimates of the most general characteristics, such as, for example, the amplitude of the leading wave. Even in this case, more accurate schemes are preferable.

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Figure 13
[79]  Pressure gauge records shown in Figure 13 demonstrate that only slight difference in the description of the final fragment of gauge record observed at point 6 results from the serious deviations in the description of processes occurring at near boundary points (1-3). The calculations using a rough grid demonstrate not only a large error in the amplitude of the leading wave, but also a clear simplification of the form of free surface oscillations, which later manifests itself in the gauge records calculated far from the coast (points 4 and 5).

2006ES000216-fig14
Figure 14
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Figure 15
[80]  The appearance of non-physical oscillations caused by insufficient resolution of the calculation grid is illustrated by the graphs shown in Figure 14 and Figure 15. It is manifested best of all if the Adams scheme is used together with a rough grid.

2006ES000216-fig16
Figure 16
[81]  The results of calculation of wave regime at the fifth point located in the coastal zone at a depth of 10 meters are shown in Figure 16. As was shown above, at this point the leading wave is practically separated from the wave reflected from the coast, and a time moment exists that divides these components of the wave field. However, it is easy to see (a) that application of rough grids ( 2h0 and h0 ) does not allow us to reproduce this phenomenon regardless increased accuracy of the Adams scheme. The graphs shown in fragment (b) evidence that the calculations with step h0 using the Adams scheme and h0/2 using the MacCormac scheme are close. A transition to the most exact grid ( h0/4 ) is shown in fragment (c). The leading wave is described ideally by each of the algorithms, while the reflected wave is reproduced best by the most accurate algorithm at the best grid. The closest version to this one is the result obtained by a more exact algorithm over a less detailed grid. Finally, for comparison, the most "exact", least "exact", and "intermediate" gauge records are shown in fragment (d). Their comparison allows us to estimate the sense of increasing the computational resources needed to increase the accuracy in the description of quantitative and qualitative characteristics of the wave regime. Similar graphs for the first and fourth gauge records are
2006ES000216-fig17
Figure 17
shown in Figure 17. These graphs indicate that the MacCormac scheme, which is easy in realization, requires doubling of calculation points (compared to the Adams scheme) in order to provide a result acceptable by accuracy. We shall consider that the acceptable reference result is the curve calculated using the Adams scheme over a grid with step h0/2. Application of more exact scheme over a smaller size grid is necessary to reproduce some "fine" effects.


RJES

Citation: Shokin, Yu. I., L. B. Chubarov, Z. I. Fedotova, S. A. Beizel, and S. V. Eletsky (2006), Principles of numerical modeling applied to the tsunami problem, Russ. J. Earth Sci., 8, ES6004, doi:10.2205/2006ES000216.

Copyright 2006 by the Russian Journal of Earth Sciences

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