Geodesic
A numerical algorithm for computing tsunami wave amplitude
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 19 (2016) no. 2, pp. 153-165.

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A numerical algorithm for computing tsunami wave front amplitude is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved by the Godunov approach and the bicharacteristic method. The qualitative comparison of the two above methods is described. Then a change in variables associated with the eikonal solution is introduced. At the last step, using the expansion of the fundamental solution of shallow water equations in the sum of singular and regular parts, we obtain the Cauchy problem for the wave amplitude. This approach allows one to reduce computer costs. The numerical results are presented. 
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S. I. Kabanikhin; O. I. Krivorotko. A numerical algorithm for computing tsunami wave amplitude. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 19 (2016) no. 2, pp. 153-165. http://geodesic.mathdoc.fr/item/SJVM_2016_19_2_a2/

[1] Voit S. S., Sebekin B. I., “Nekotorye gidrodinamicheskie modeli neustanovivshikhsya volnovykh dvizhenii tipa voln tsunami”, Morskie gidrofizicheskie issledovaniya, 1, MGI AN USSR, Sevastopol, 1968, 137–145

[2] Babich V. M., Buldyrev V. S., Molotkov I. A., Prostranstvenno-vremennoi luchevoi metod: lineinye i nelineinye volny, Izd-vo Leningradskogo universiteta, L., 1985 | MR

[3] Dobrokhotov S. Yu., Nekrasov R. V., Tirozzi B., “Asymptotic solutions of the linear shallow-water equations with localized initial data”, J. of Engineering Mathematics, 69 (2011), 225–242 | DOI | MR | Zbl

[4] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Moskovskogo universiteta, M., 1965

[5] Marchuk An. G., Vasiliev G. S., “The fast method for a rough tsunami amplitude estimation”, Bull. Novosibirsk Comp. Center. Ser. Math. Model. in Geoph., 17, Novosibirsk, 2014, 21–34

[6] Kosykh V. S., Chubarov L. B., Gusyakov V. K., Kamaev D. A., Grigoreva V. M., Beizel S. A., “Metodika rascheta maksimalnykh vysot voln tsunami v zaschischaemykh punktakh poberezhya Dalnego vostoka Rossiiskoi federatsii”, Rezultaty ispytaniya novykh i usovershenstvovannykh tekhnologii, modelei i metodov gidrometeorologicheskikh prognozov, 40, 2013, 115–134

[7] Airy G., “Tides and Waves”, Encyclopedia Metropolitana, 5 (1845), 241–396

[8] Kabanikhin S. I., Krivorot'ko O. I., “A numerical method for determining the amplitude of a wave edge in shallow water approximation”, Applied and Computational Mathematics, 12:1 (2013), 91–96 | MR | Zbl

[9] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1981 | MR

[10] Kabanikhin S. I., Lineinaya regulyarizatsiya mnogomernykh obratnykh zadach dlya giperbolicheskikh uravnenii, Preprint No 27, AN SSSR. Sib.otd-nie. Institut matematiki, Novosibirsk, 1988

[11] Petrashen G. I., Rasprostranenie volnovykh polei signalnogo tipa v uprugikh seismicheskikh sredakh, Izd-vo Sankt-Peterburgskogo universiteta, SPb., 2000

[12] Arnold V. I., Osobennosti kaustik i volnovykh frontov, FAZIS, M., 1996 | MR

[13] Zeldovich B. Ya., Pilipetskii N. F., Shkunov V. V., Obraschenie volnovogo fronta, Nauka, M., 1985

[14] Romanov V. G., Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984 | MR

[15] Bezhaev A. Yu., Lavrentiev M. M. (jr.), Marchuk An. G., Titov V. V., “Determination of tsunami sources using deep ocean wave records”, Bull. Novosibirsk Comp. Center. Ser. Math. Model. in Geoph., 11, Novosibirsk, 2006, 53–62

[16] Kabanikhin S. I., Krivorotko O. I., “Chislennoe reshenie uravneniya eikonala”, Trudy IV mezhdunarodnoi molodezhnoi shkoly-konferentsii “Teoriya i chislennye metody resheniya obratnykh i nekorrektnykh zadach”. Chast I, Sibirskie elektronnye matematicheskie izvestiya, 10 (2013), S.28–S.34

[17] Godunov S. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matematicheskii sbornik, 47(89):3 (1959), 271–306 | MR | Zbl

[18] Ivanov D. I., Ivanov I. E., Kryukov I. A., “Algoritmy priblizhennogo resheniya nekotorykh zadach prikladnoi geometrii, osnovannye na uravnenii tipa Gamiltona–Yakobi”, Zhurn. vychisl. matem. i mat. fiziki, 45:8 (2005), 1345–1358 | MR | Zbl

[19] Elsgolts L. E., Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka, M., 1969

[20] Romanov V. G., Ustoichivost v obratnykh zadachakh, Nauchnyi mir, M., 2005 | MR

[21] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sibirskoe nauchnoe izd-vo, Novosibirsk, 2009

[22] Khakimzyanov G. S., Shokin Yu. I., Barakhnin V. B., Shokina N. Yu., Chislennoe modelirovanie techenii zhidkosti s poverkhnostnymi volnami, Izd-vo SO RAN, Novosibirsk, 2001