RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES6004, doi:10.2205/2006ES000216, 2006
[82] A comparison analysis of two programming systems (programming code packages) is an absolutely needed stage preceding the formulation of numerical modeling program. Here, we shall consider a specific example of such comparison and suggest a set of test problems for discussion, each of which is designed for determining and estimating specific characteristics of different models, algorithms, and program realizations.
[83] The main attention would be focused on the Nereus programming system developed by the authors and designed for numerical modeling of tsunami propagation from the source to the coast including wave interaction with island systems, coastal constructions, and flooding of the adjacent land [Eletsky et al., 2005].
[84] The capabilities of this system are compared with the capabilities of the program developed by specialists from Nizhniy Novgorod and Turkey on the basis of the TUNAMI code [Goto et al., 1997; Zaitsev et al., 2005] in its turn developed by Japan specialists in the 1980s. The recent modifications of this code claim to be a standard, thus they are widely spread in the community of the specialists involved in the solution of applied tsunami problems. Both systems are based on classical equations of the theory of shallow water. The algorithms realized by the systems are related to the class of explicit finite difference schemes. The Nereus and TUNAMI programs to one or another degree of flexibility allow us to perform calculations with or without account for nonlinear effects, rotation of the Earth, bottom and wind friction.
[85] The modeling results of Sumatra (2004) tsunami in the Indian Ocean became the material for comparison. The data obtained using one of the latest versions of TUNAMI system (we shall call it TUNAMI-M) were kindly given to us by E. N. Pelinovsky and A. I. Zaitsev, the active participants of the modification process and development of this code
[86] A number of tests were suggested by the authors to estimate the efficiency of programming systems. In all problems considered below, condition of zero transport (reflection) is specified along the coastline, which corresponds to the assumption about the existence of a vertical wall at the "land-sea" boundary. Thus, runup of waves is not considered in these problems.
[87] As was mentioned above, one of the main problems of numerical modeling of tsunami is reproducing the coastal wave regimes. In this relation, the problem about the quality of reproducing the boundary conditions at the coastlines, whose configuration complexity is close to the real ones, becomes especially important. This problem becomes the first, since the methods, which were used as the basis of many specialized (tsunami oriented) computational algorithms including the algorithms of the Nereus system, use uniform rectangular grids. Thus, the first test problems are directly related to the verification of the efficiency of numerical methods in the description of the interaction between waves and coastal structures of arbitrary form.
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Figure 18 |
[89] Here and above, we suggest using mainly gauge records to estimate the solution and perform comparative analysis. The modeling area is specified by a rectangle with sides 3267,000 m (in the direction of the wave propagation) and 2913,000 m. The corresponding computational grid has a size of 1090 by 972 points. The depth of the basin is constant equal to 1000 m. The vertical wall is located in the center of the modeling area. The front of the solitary wave with amplitude of 1 m is parallel to the wall. At the initial time moment, the wave crest is located over a point with coordinate 2451,000 m. Initial velocity field is specified so that the wave propagates without changing its form in the direction to the wall. Reflection conditions are specified at the lateral boundaries of the calculation area along the wave propagation, while at the back boundary free pass condition is specified. The physical time of the process is 24,000 seconds.
[90] Inclusion of a significant fragment consisting of land points, in which the calculations are not performed, is explained by the idea to conserve unique geometrical characteristics of the test problems.
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Figure 19 |
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Figure 20 |
[93] The next series of the test problems is directed to distinguish the efficiency and principal possibility of using the test program systems for numerical modeling of tsunami. These are the Nereus and TUNAMI-M programs.
[94] Problems 4-6 are related to a certain degree to the modeling of the catastrophic Sumatra tsunami (26 December 2004).
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Figure 21 |
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Figure 22 |
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Figure 23 |
[98] The next series of figures presents the most general characteristics of wave regimes corresponding to the test problems considered above. No doubt, the results presented here are of purely illustrative character. A number of steps organizing testing and certification of programming systems designed for numerical modeling of tsunami should be made if serious work is planned. The materials should be prepared to use in the test programs including electronic presentation of the data in agreed formats. These materials would contain all the necessary input data and full results including the wave and velocity fields calculated at given time moments and gauge records at specified points.
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Figure 24 |
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Figure 25 |
[101] As the wave approaches the island, it begins to interact with it, turns around the island and forms a system of concentrated reflected waves, which propagate in the opposite direction. The front of the wave after insignificant changes restores its characteristics and continues the motion in the initial direction, while a configuration is formed behind the front frequently called a "dove tail". In addition to these qualitative results, important material for estimating the computational tools can be obtained from the analysis of the distribution of the maximal and minimal wave heights along the perimeter of the island. These results also facilitate the estimate of the quality of reflecting boundary conditions and determination whether the resolution of the computational grid is sufficient.
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Figure 26 |
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Figure 27 |
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Figure 28 |
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Figure 29 |
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Figure 30 |
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Figure 31 |
[104] As to the comparison of gauge records calculated using two different programming systems, it is reasonable to start the comparison from the simplest problem 6 (in the absence of variations in the depths and coastal boundaries). On the basis of the analysis of gauge records shown in Figure 27 we can speak not only about the qualitative but also about quantitative coincidence of the modeling results. We note that different conditions at the boundaries of the computational area are used in the calculations presented here. This is clearly seen in the record of gauge 15. Natural condition of free pass of the wave was imposed at the southern boundary in the calculations using the Nereus program, while in the TUNAMI-M programming system another boundary condition was specified.
[105] The comparison of the results of the solution of more complex problem 5 (see Figure 28), in which the real form of the coastal boundaries is conserved, while the depth remains constant, demonstrates that the deviation of the results becomes more notable, however, the leading waves are reproduced almost identically. In some cases (not included in the material described here) we can speak about complete qualitative coincidence of the results. The cause of the noted differences can be a risky joint use of linear and nonlinear models of shallow water in the algorithm of the TUNAMI-M program, which requires for the correct realization a thorough sewing the solutions with account for the difference in the directions of characteristics used for transition of solutions in hyperbolic equations. The Nereus programming system uses only nonlinear theory of shallow water.
[106] Finally, the results of modeling of problem 4 (see Figure 29), which is the closest to the real tsunami phenomenon, indicate that regardless that the leading waves are described almost identically, the results diverge stronger and stronger in the course of time.
[107] The analysis of the entire set of the materials of calculations (no doubt, there are points, in which solutions differ not so strongly) leads to a conclusion that the cause of these differences is most likely the different approaches of the authors of programming systems to testing of algorithms both at internal and boundary points. It is our opinion that distinguishing of such differences should not be limited only by stating of this fact. We should apply all forces to understand their cause and eliminate the error in algorithms and programs if there are any. Test problems 5 and 6 were suggested with this goal in mind. The analysis of their solutions allowed us to determine the possible means of solving this crisis. We note that such work requires certain enthusiasm and personal will for the cooperation between the developers of the models, algorithms, and programs, as well as organization forcing on behalf of the entire community of the specialists.
[108] Concluding the description of the results of numerical solution of problems 4-6 we shall briefly describe the above mentioned analysis of the influence of real bottom topography, indentation of the coastal boundaries, and combination of these factors on the solution. Pressure gauges presented in Figure 30 and Figure 31 are the illustrations of the perspectives of such analysis using the materials of calculations. The presence of high frequency oscillations in the results obtained using the TUNAMI-M programming system is worth attention. The appearance of such oscillations in problem 6 with the bottom of constant depth and absence of internal boundaries points to the purely computational nature of such oscillations and impossibility of associating them with any physical sense.
[109] The further interaction of the "numerical" oscillations with irregularities of bottom topography and peculiarities of the coastal boundaries can become a cause of increasing distortion of the solution of more complex problems 4 and 5 and increasing discrepancy with the results of calculations using another programming system. In such cases, one should seriously treat the reveake problem and make efforts to overcome it.
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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