Dependence of the computed tsunami wave heights on the grid resolution

M. M. Lavrentiev ; K. F. Lysakov ; An. G. Marchuk ; K. K. Oblaukhov ; M. Yu. Shadrin

Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 2, pp. 327-342.

Résumé

Tsunami after the March 11, 2011, as well as the other recent events, have shown that destructive tsunami waves generated by earthquakes continue to pose a significant risk to coastal populations adjacent to subduction zones, where most of tsunami sources are located. In some places along these coasts, the tsunami run-up heights can reach 30 m or more, causing destruction and casualties. However, the wave heights maxima are distributed very nonuniformly along the coast with sharp local peaks in amplitude. Since for near-shore events the tsunami wave arrival time at the nearest coastal point after an earthquake is on the order of 20 minutes, a quick (within 1-2 minutes) correct assessment of the distribution of maximum wave heights along the coast will allow warning services take evacuation actions exactly where needed. Modern modelling tools allowing quickly calculate wave parameters with sufficient accuracy if the wave characteristics at the initial time instance are known. However, this requires calculations in spatial steps of several meters, which is time-consuming even when using supercomputers. In addition, in the case of a strong earthquake, power outages are possible, which does not guarantee that numerical modelling can be started immediately after the seismic event. The use of large, hundreds of meters resolution calculation grid does not allow estimate correctly the tsunami wave heights near the shore. Fine grids entail the growth of the duration of computing time. The resolution of this contradiction dictates the necessity to choose the optimal correlation between grid spacing (results precision) and calculation time. In this paper the dependence of the calculated tsunami wave parameters depending on the grid spacing is studied. Obtained results will be used for optimal selection of application zones of meshes with different spacing. Computational experiments were carried out on a personal computer (PC) using hardware acceleration – a specialized FPGA-based microchip (FPGA being Field Programmable Gates Array), used with the computer as a coprocessor. As a result, a sufficiently high performance of calculations is achieved. Calculation of wave parameters near the shore on the computational grid of 3000$\times$2500 nodes takes less than 1 min. In addition, the proposed solution does not depend on possible power supply failures.

Détail

Keywords: system of shallow water equations, Mac-Cormack numerical scheme, tsunami wave height.

Bibliographie

[1] Giga E., Spillane M., Titov V., Chamberlin C., Newman J., Development of the forecast propagation database for NOAA's short-term inundation forecast for tsunamis (SIFT), NOAA Tech. Memo. OAR PMEL-139, NOAA, Washington, 2008

[2] Kowalik Z., Murty T. S., Numerical modeling of ocean dynamics, World Scientific, Singapore, 1993

[3] Lavrentiev M., Lysakov K., Marchuk An., Oblaukhov K., “Fundamentals of fast tsunami wave parameter determination technology for hazard mitigation”, Sensors, 22 (2022), 7630 | DOI

[4] Lavrentiev M., Lysakov K., Marchuk An., Oblaukhov K., Shadrin M., “Hardware acceleration of tsunami wave propagation modeling in the southern part of Japan”, Appl. Sci., 10 (2020), 4159 | DOI

[5] Lavrentiev M. M., Marchuk An. G., Oblaukhov K. K., Romanenko A. A., “Comparative testing of MOST and Mac-Cormack numerical schemes to calculate tsunami wave propagation”, J. Phys. Conf. Ser., 1666 (2020), 012028 | DOI

[6] Lax P. D., Richtmyer R. D., “Survey of the stability of linear finite difference equations”, Commun. Pure Appl. Math., 9 (1956), 267–293 | DOI | MR | Zbl

[7] Liang Q., Hou J., Amouzgar R., “Simulation of tsunami propagation using adaptive cartesian grids”, Coast. Engrg. J., 57:4 (2015), 1550016-1–1550016-30

[8] MacCormack R. W., Paullay A. J., “Computational efficiency achieved by time splitting of finite-difference operators”, AIAA Meeting Paper. 10th Aerospace Sciences Meeting, San-Diego, 1972, 154

[9] Popinet S., “Quadtree-adaptive tsunami modelling”, Ocean Dynamics, 61 (2011), 1261–1285 | DOI

[10] Shuto N., Goto C., Imamura F., “Numerical simulation as a means of warning for near field tsunamis”, Coast. Eng. Jpn., 33 (1990), 173–193 | DOI

[11] Stoker J. J., Water waves. The mathematical theory with applications, Interscience Publ., New York, 1957 | MR | Zbl

[12] Titov V. V., Gonzalez F. I., Implementation and testing of the method of splitting tsunami (MOST) model, NOAA Tech. Memo. ERL PMEL-112, NOAA, Washington, 1997

[13] Wang X., Power W. L., COMCOT: a tsunami generation propagation and run-up model, GNS Science, Lower Hutt City, 2011

[14] Yalciner A. C., Alpar B., Altinok Y., Ozbay I., Imamura F., “Tsunamis in the sea of Marmara: historical documents for the past, models for future”, Mar. Geol., 190 (2002), 445–463 | DOI

[15] Zaytsev A., Kurkin A., Pelinovsky E., Yalciner A. C., “Numerical tsunami model NAMI_DANCE”, Sci. Tsunami Hazards, 38 (2019), 151–168

[16] 500m Gridded Bathymetry Data, Japan Oceanographic Data Center (data obrascheniya: 26.06.2024) https://www.jodc.go.jp/jodcweb/JDOSS/infoJEGG.html

[17] 15 Arc-Second Global Relief Model, NOAA National Centers for Environmental Information, 2022 (data obrascheniya: 26.06.2024) https://www.ngdc.noaa.gov/mgg/global/relief/ETOPO2022/data/15s/15s_surface_elev_gtif/

[18] Natural disasters in aused record economic losses, EarthSky (data obrascheniya: 21.06.2024) http://earthsky.org/earth/economic-losses-from-earthquakes-and-natural-disasters-peaked-in-2011

[19] Vitis High-Level Synthesis User Guide, AMD Technical Information Portal (data obrascheniya: 21.06.2024) https://docs.amd.com/r/en-US/ug1399-vitis-hls