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On the Properties of Difference Schemes for Solving Nonlinear Dispersive Equations of Increased Accuracy. I. The Case of One Spatial Variable

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Abstract

A difference scheme of the predictor–corrector type is constructed for solving nonlinear dispersive equations of wave hydrodynamics with an increased order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.

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REFERENCES

  1. Kabanikhin, S.I. and Krivorotko, O.I., A Numerical Algorithm for Computing Tsunami Wave Amplitude, Num. An. Appl., 2016, vol. 9, no. 2, pp. 118–128.

    Article  MathSciNet  Google Scholar 

  2. Hayshi, K., Marchuk, An., and Vazhenin, A.P., Generating Boundary Conditions for the Calculation of Tsunami Propagation on Nested Grids, Num. An. Appl., 2018, vol. 11, no. 3, pp. 256–267.

    Article  MathSciNet  Google Scholar 

  3. Serre, F., Contribution à L’étude des Écoulements Permanents et Variables Dans Les Canaux, La Houille Blanche, 1953, vol. 3, pp. 374–388.

    Article  Google Scholar 

  4. Green, A.E. and Naghdi, P.M., A Derivation of Equations for Wave Propagation in Water of Variable Depth, J. Fluid Mech., 1976, vol. 78, pp. 237–246.

    Article  MATH  Google Scholar 

  5. Zheleznyak, M.I. and Pelinovsky, E.N., Physical-Mathematical Models of Tsunami Run-Up on Shore, Nakat tsunami na bereg. Sb. nauchn. trudov (Tsunami Run-Up on Shore), Gorky: Institute of Applied Physics, USSR Acad. Sci., 1985, pp. 8–33.

  6. Khakimzyanov, G., Dutykh, D., Fedotova, Z., and Gusev, O., Dispersive Shallow Water Waves: Theory, Modeling, and Numerical Methods, Basel: Birkhäuser, 2020.

    Book  MATH  Google Scholar 

  7. Madsen, P.A., and Schäffer, H.A., Higher Order Boussinesq-Type Equations for Surface Gravity Waves: Derivation and Analysis, Philos. Trans. Royal Soc. London. Ser. A, 1998, vol. 356, pp. 3123–3181.

    Article  MathSciNet  MATH  Google Scholar 

  8. Khakimzyanov, G.S., Fedotova, Z.I., and Dutykh, D., Two-Dimensional Model of Wave Hydrodynamics with High Accuracy Dispersion Relation, Vych. Tekhnol., 2020, vol. 25, no. 5, pp. 17–41.

    Google Scholar 

  9. Benjamin, T.B., Bona, J.L., and Mahony, J.J., Model Equations for Long Waves in Nonlinear Dispersive Systems, Philos. Trans. Royal Soc. London. Ser. A, 1972, vol. 272, pp. 47–78.

    MathSciNet  MATH  Google Scholar 

  10. Fedotova, Z.I., Khakimzyanov, G.S., and Gusev, O.I., History of the Development and Analysis of Numerical Methods for Solving Nonlinear Dispersive Equations of Hydrodynamics. I. One-Dimensional Models Problems, Vych. Tekhnol., 2015, vol. 20, no. 5, pp. 120–156.

    MATH  Google Scholar 

  11. Brocchini, M., A Reasoned Overview on Boussinesq-Type Models: The Interplay between Physics, Mathematics and Numerics, Proc. Royal Soc. A, 2013, vol. 469, ID-number 20130496; DOI: 10.1098/ rspa.2013.0496.

    Article  MathSciNet  MATH  Google Scholar 

  12. Madsen, P.A., Murray, R., and Sorensen, O.R., A New Form of the Boussinesq Equations with Improved Linear Dispersion Characteristics, Coastal Engin., 1991, vol. 15, pp. 371–388.

    Article  Google Scholar 

  13. Voltsinger, N.E. and Pyaskovskii, R.V., Teoriya melkoi vody. Okeanologicheskie zadachi i chislennye metody (Shallow Water Theory. Oceanology Problems and Numerical Methods), Leningrad: Gidrometeoizdat, 1977.

    Google Scholar 

  14. Belikov, V.V. and Semenov, A.Yu. A Godunov-Type Method Based on an Exact Solution to the Riemann Problem for the Shallow-Water Equations, Comput. Math. Math. Phys., 1997, vol. 37, no. 8, pp. 974–986.

    MATH  Google Scholar 

  15. Fedotova, Z.I. and Khakimzyanov, G.S., Characteristics of Finite Difference Methods for Dispersive Shallow Water Equations, Russ. J. Num. An. Math. Model., 2016, vol. 31, no. 3, pp. 149–158.

    Article  MathSciNet  MATH  Google Scholar 

  16. Fedotova, Z.I., Khakimzyanov, G.S., Gusev, O.I., and Shokina, N.Y., History of the Development and Analysis of Numerical Methods for Solving Nonlinear Dispersive Equations of Hydrodynamics. II. Two-Dimensional Models, Vych. Tekhnol., 2017, vol. 22, no. 5, pp. 73–109.

    MATH  Google Scholar 

  17. Khakimzyanov, G.S., Fedotova, Z.I., Gusev, O., and Shokina, N.Yu., Finite Difference Methods for 2D Shallow Water Equations with Dispersion, Russ. J. Num. An. Math. Model., 2019, vol. 34, no. 2, pp. 105–117.

    Article  MathSciNet  MATH  Google Scholar 

  18. Gusev, O.I., Shokina, N.Y., Kutergin, V.A., and Khakimzyanov, G.S., Numerical Modelling of Surface Waves Generated by Underwater Landslide in a Reservoir, Vych. Tekhnol., 2013, vol. 18, no. 5, pp. 74–90.

    Google Scholar 

  19. Shokin, Yu.I., Metod differentsial’nogo priblizheniya (The Method of Differential Approximation), Novosibirsk: Nauka, 1979.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Federal Research Center for Information and Computational Technologies, Novosibirsk, Russia

    Z. I. Fedotova & G. S. Khakimzyanov

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  1. Z. I. Fedotova
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  2. G. S. Khakimzyanov
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Correspondence to Z. I. Fedotova or G. S. Khakimzyanov.

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Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2023, Vol. 26, No. 4, pp. 451-467. https://doi.org/10.15372/SJNM20230408.

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Fedotova, Z.I., Khakimzyanov, G.S. On the Properties of Difference Schemes for Solving Nonlinear Dispersive Equations of Increased Accuracy. I. The Case of One Spatial Variable. Numer. Analys. Appl. 16, 375–389 (2023). https://doi.org/10.1134/S1995423923040080

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  • DOI: https://doi.org/10.1134/S1995423923040080

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