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Article information
2015 , Volume 20, ¹ 5, p.120-156
Fedotova Z.I., Khakimzyanov G.S., Gusev O.I.
History of the development and analysis of numerical methods for solving nonlinear dispersive equations of hydrodynamics. I. One-dimensional models problems
The article describes the main stages of development of finite-difference methods for numerical solving of nonlinear dispersive (NLD) hydrodynamic equations and presents new results on the theoretical research. The peculiarity of NLD-equations is a presence of mixed derivatives of the third order in time and space, which bring special features into difference schemes used for their approximation. This affects their theoretical properties (stability conditions, balancing in modeling of numerical and physical dispersion, etc.) and well as the ways for implementation of numerical algorithms. The paper has analyzed four groups of finite difference algorithms. The first one is the schemes, for which the discrete models are designed by direct approximation of all members of equations, including the mixed derivatives of the third order, with scalar sweeps for realization of numerical algorithm. The second group includes those numerical algorithms that are based on the decomposition of the NLD-equations into a system of ordinary differential equations and into the differential equations that do not contain time derivatives. To the third group we refer the algorithms which use special schemes of higher order accuracy in the methods of the first and second groups. Finally, the last group consists of difference schemes based on the splitting of the NLD-equations in such a way that there is the scalar equation of an elliptic type for the dispersive component of the pressure and there is a hyperbolic system of equations, which mimics the nondispersive shallow water model. Since in the full statement the NLD-equations and their finite-difference approximation are not amenable to analytical study, it is necessary to simplify formulations for the study of properties in particular cases. Here are considered the dispersive scalar equations and systems of linear analogues of the NLD-equations in the cases of both a plane and other profiles of the bottom. Such studies help to clarify the essence of numerical methods for the NLD-models and highlight the advantages and disadvantages of methods. They also show their distinction from methods of solving nondispersive shallow water equations. In the process of this study the corrected conditions of stability and new knowledge on the dispersion properties of finite-difference schemes are obtained. The conditions of stability of schemes for dispersive equations turned out to be different than for hyperbolic ones. The difference is that the specified conditions of stability includes a new parameter characterizing the fineness of the grid compared to the characteristic depth, and in the limit of grid refinement the new conditions are formulated in the form of restrictions on the time step only. It was shown that in the area of stability of some schemes, there are the values of the Courant number, for which the influence of “scheme dispersion” is minimal. Difference schemes for which the approximation error of dispersion is decreased only by grinding of the computational grid are identified.
[full text]
Keywords: nonlinear dispersive equations, numerical algorithms, finite-difference methods, accuracy, stability, dissipation, dispersion
Author(s):
Fedotova Zinaida Ivanovna
PhD.
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Lavrentiev ave. 6
Phone Office: (383) 334-91-21
E-mail: zf@ict.nsc.ru
Khakimzyanov Gayaz Salimovich
Dr. , Professor
Position: Leading research officer
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: khak@ict.nsc.ru
SPIN-code: 3144-0877Gusev Oleg Igorevitch
PhD.
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, 6, Acad. Lavrentjev avenue
Phone Office: (383) 334-91-18
E-mail: GusevOI@ict.sbras.ru
SPIN-code: 3995-2134 References:
[1] Mei, C.C., Le Mehaute, B. Note on the equations of long waves over an uneven bottom. Journal of Geophysical Research. 1966; 71(2):393–400.
[2] Peregrine, D.H. Long waves on a beach. Journal of Fluid Mechanics. 1967; (27):815–827.
[3] Rozhdestvenskiy, B.L., Yanenko, N.N. Sistemy kvazilineynykh uravneniy i ikh prilozheniya k gazovoy dinamike [Systems of quasilinear equations and their application to gas dynamics]. Moscow: Nauka; 1968: 592. (In Russ.)
[4] Shokin, Yu.I., Yanenko N.N. Metod differentsial'nogo priblizheniya. Primenenie k gazovoy dinamike. Novosibirsk: Nauka, Sibirskoe otdelenie; 1985: 364. (In Russ.)
[5] Fedotova, Z.I., Khakimzyanov, G.S. The basic nonlinear-dispersive hydrodynamic model of long surface waves. Computational Technologies. 2014; 19(6):77–94. (In Russ.)
[6] Shokin, Yu.I., Fedotova, Z.I., Khakimzyanov, G.S. Hierarchy of nonlinear models of the hydrodynamics of long surface waves. Doklady Physics. 2015; 60(5):224–228.
[7] Ertekin, R.C., Webster, W.C., Wehausen, J.V. Waves caused by a moving disturbance in a shallow channel of finite width. Journal of Fluid Mechanics. 1986; (169):275–292.
[8] Zheleznyak, M.I., Pelinovskiy, E.N. Fiziko-matematicheskie modeli nakata tsunami na bereg. Nakat tsunami na bereg: Sbornik nauchykh. trudov [Physico-mathematical models of the tsunami climbing a beach. Tsunami climbing a beach: Collection of scientific papers]. Gorky: Institute Applied Physics Press; 1985: 8–33. (In Russ.)
[9] Fedotova, Z.I., Khakimzyanov, G.S. Nonlinear dispersive shallow water equations for a non-stationary bottom. Computational Technologies. 2008; 13(4):114–126. (In Russ.)
[10] Fedotova, Z.I., Khakimzyanov, G.S., Dutykh, D. On the energy equation of approximate models in the long-wave hydrodynamics. Russian Journal of Numerical Analysis and Mathematical Modelling. 2014; 29(3):167–178.
[11] Su, C.H., Gardner, C.S. Korteweg — de Vries equation and generalizations. III. Derivation of the Korteweg — de Vries equation and Burgers equation. Journal of Mathematical Physics. 1969; 10(3):536–539.
[12] Peregrine, D.H. Calculations of the development of an undular bore. Journal of Fluid Mechanics. 1966; 25(2):321–331.
[13] Benjamin, T.B., Bona, J.L., Mahony, J.J. Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. A. 1972; (272):47–78.
[14] Whitham, G.B. Linear and nonlinear waves. N.Y.: John Wiley & Sons Inc.; 1974: 636.
[15] Pelinovskiy, E.N. Gidrodinamika voln tsunami. [Tsunami Wave Hydrodynamics]. Nizhniy Novgorod: Institute Applied Physics Press; 1996: 276. (In Russ.)
[16] Olver, P.J. Euler operators and conservation laws of the BBM equation. Mathematical Proceedings of the Cambridge Philosophical Society. 1979; (85):143–160.
[17] Hereman, W. Shallow water waves and solitary waves. Mathematics of Complexity and Dynamical Systems (Ed. R.A. Meyers). N.Y.: Springer; 2011:1520–1532.
[18] Novikov, V.A., Fedotova, Z.I. Numerical modelling of long wave propagation in bays on the base of simplified Boussinesq model. Collect. Sientific Papers, Proceedings of All-union Conference on Numerical Methods in Wave Hudrodynamics Problems 23–25 Sept., 1990, Rostov-na-Dony. Krasnoyarsk: Computer Center of the Siberian Branch of Academy of Sciences of the USSR; 1991:9–14. (In Russ.)
[19] Fedotova, Z.I. On properties of two simplified nonlinearly dispersive models of long wave hydrodynamics. International Journal of Computational Fluid Dynamics. 1998; 10(2):159–171.
[20] Khabakhpashev, G.A. Nonlinear evolition equation for moderately long two-dimensional waves on a free surface of viscous liquid. Computational Technologies. 1997; 2(2):94–101. (In Russ.)
[21] Litvinenko, A.A., Khabakhpashev, G.A. Numerical modeling of sufficiently long nonlinear two-dimensional waves on water surface in a basin with a gently sloping bottom. Computational Technologies. 1999; 4(3):95–105. (In Russ.)
[22] Kim, K.Y., Reid, R.O., Whitaker, R.E. On an open radiational boundary condition for weakly dispersive tsunami waves. Journal of Computational Physics. 1988; (76):327–348.
[23] Bogolyubskiy, I.L. Modified equation of a nonlinear string and inelastic interaction of solitons. Pis'ma v Zhurnal eksperimental'noy i teoreticheskoy fiziki. 1976; 24(3):184–186. (In Russ.)
[24] Khabakhpashev, G.A. Effect of bottom friction on the dynamics of gravity perturbations. Fluid Dynamics. 1987; 22(3):430–437.
[25] Richtmyer, R.D., Morton, K.W. Difference methods for initial-value problems. N.Y.: Interscience Publishers; 1967: 405.
[26] Eilbeck, J.C., McGuire, G.R. Numerical study of the regularized long-wave equations 1: Numerical methods. Journal of Computational Physics. 1975; (19):43–57.
[27] Samarskii, A.A. The theory of difference schemes. USA: Marcel Dekker; 2001: 788.
[28] Kompaniets, L.A. Analysis of difference algoritms for nonlinear dispersive shallow water models. Russian Journal of Numerical Analysis and Mathematical Modelling. 1996; 11(3):205–222.
[29] Fedotova, Z.I., Pashkova, V.Yu. Methods of construction and the analysis of difference schemes for nonlinear dispersive models of wave hydrodynamics. Russian Journal of Numerical Analysis and Mathematical Modelling. 1997; 12(2):127–149.
[30] Chubarov, L.B., Fedotova, Z.I., Shokin, Yu.I., Einarsson, B.G. Comparative analysis of nonlinear dispersive models of shallow water. International Journal of Computational Fluid Dynamics. 2000; 14(1):55–73.
[31] Godunov, S.K., Ryabenkii, V.S. Difference schemes. Translation by E.M. Gelbard. Amsterdam: North-Holland; 1987: 486.
[32] LeVeque, R.J. Numerical methods for conservation laws. Berlin: Birkhauser Verlag; 1992: 214.
[33] Dutykh, D., Clamond, D., Milewski, P., Mitsotakis, D. Finite volume and pseudospectral schemes for the fully nonlinear 1D Serre equatios. European Journal of Applied Mathematics. 2013; 24(5):761–787.
[34] Grue, J., Pelinovsky, E.N., Fructus, D., Talipova, T., Kharif, C. Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami. Journal of Geophysical Research. 2008; 113. C05008.
[35] Glimsdal, S., Pedersen, G.K., Atakan, K., Harbitz, C.B., Langtangen, H.P., Lovholt, F. Propagation of the Dec. 26, 2004, Indian Ocean Tsunami: Effects of dispersion and source characteristics. International Journal of Fluid Mechanics Research. 2006; 33(1):15–43.
[36] Shokin, Yu.I., Chubarov, L.B. The numerical modelling of long wave propagation in the framework of non-linear dispersion models. Computers and Fluids. 1987; 15(3):229–249.
[37] Abbott, M.B., Petersen, H.M., Skovgaard, O. On the numerical modelling of short waves in shallow water. Journal of Hydraulic Research. 1978; 16(3):173–204.
[38] Abbott, M.B., McCowan, A.D., Warren, I.R. Accuracy of short-wave numerical models. Journal of Hydraulic Engineering. 1984; 110(10):1287–1301.
[39] Rygg, O.B. Nonlinearv refraction-diffraction of surface waves in intermediate and shallow water. Coastal Engineering Journal. 1988; 12(3):191–211.
[40] Chubarov, L.B., Fedotova, Z.I., Shkuropatskii, D.A. Investigation of computational models of long surface waves in the problem of interaction of a solitary wave with a conic island. Russian Journal of Numerical Analysis and Mathematical Modelling. 1998; 13(4):289–306.
[41] Bonneton, P., Barthelemy, E., Chazel, F., Cienfuegos, R., Lannes, D., Marche, F., Tisser, M. Recent advances in Serre — Green — Naghdi modelling for wave transformation, breaking and runup processes. European Journal of Mechanics — B/ Fluids. 2011; (30):589–597.
[42] Kompaniets, L.A. On difference schemes for nonlinear dispersive Green — Naghdi and Aleshkov models with improved approximation of dispersion relations. Computational Technologies. 1995; 4(11):144–153. (In Russ.)
[43] Nwogu, O. Alternative form of Boussinesq equations for nearshore wave propagation. Journal of Waterway, Port, Coastal, and Ocean Engineering. 1993; 119(6):618–638.
[44] Wei, G., Kirby, J.T. Time-dependent numerical code for extended Boussinesq equations. Journal of Waterway, Port, Coastal, and Ocean Engineering. 1995; 121(5):251–261.
[45] Barakhnin, V.B., Khakimzyanov, G.S. On the algorithm for one nonlinear dispersive shallow-water model. Russian Journal of Numerical Analysis and Mathematical Modelling. 1997; 12(4):293–317.
[46] Ramos, J.I. Explicit finite difference method for the EW and RLW equation. Applied Mathematics and Computation. 2006; (179):622–638.
[47] Walkley, M., Berzins, M. A finite element method for the one-dimensional extended Boussinesq equations. International Journal for Numerical Methods in Fluids. 1999; (29):143–157.
[48] Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T. A high-order adaptive timestepping TVD solver for Boussinesq modelling of breaking waves and coastal inundation. Ocean Modelling. 2012; (43–44):36–51.
[49] Kirby, J.T., Shi, F., Tehranirad, B., Harris, J.C., Grilli, S.T. Dispersive tsunami waves in the ocean: Model equations and sensitivity to dispersion and Coriolis effects. Ocean Modelling. 2013; (62):39–55.
[50] Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. Journal of Fluid Mechanics. 1995; (294):71–92.
[51] Cienfuegos, R., Barthelemy, E., Bonneton, P. A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. International Journal for Numerical Methods in Fluids. 2006; (51):1217–1253.
[52] Gusev, O.I., Shokina, N.Yu., Kutergin, V.A., Khakimzyanov, G.S. Numerical modelling of surface waves generated by underwater landslide in a reservoir. Computational Technologies. 2013; 18(5):74–90. (In Russ.)
[53] Gusev, O.I. Algorithm for surface waves calculation above a movable bottom within the frame of plane nonlinear dispersive model. Computational Technologies. 2014; 19(6):19–40. (In Russ.)
[54] Shokin, Yu.I., Beisel, S.A., Gusev, O.I., Khakimzyanov, G.S., Chubarov, L.B., Shokina, N.Yu. Numerical modelling of dispersive waves generated by landslide motion. Bulletin of the South Ural State University. 2014; 7(1):121–133. (In Russ.)
[55] Khakimzyanov, G.S., Gusev, O.I., Beisel, S.A., Chubarov, L.B., Shokina, N.Yu. Simulation of tsunami waves generated by submarine landslides in the Black Sea. Russian Journal of Numerical Analysis and Mathematical Modelling. 2015; 30(4):227–237.
[56] Gusev, O.I., Khakimzyanov, G.S. Numerical simulation of long surface waves on a rotating sphere within the framework of the full nonlinear dispersive model. Computational Technologies. 2015; 20(3):3–32. (In Russ.)
[57] Barakhnin, V.B., Khakimzyanov, G.S. The spliting technique as applied to the solution of the nonlinear dispersive shallow-water equations. Doklady Mathematics. 1999; 59(1):70–72.
[58] Khakimzyanov, G.S., Shokin, Yu.I., Barakhnin, V.B., Shokina, N.Yu. Chislennoe mode-lirovanie techenij zhidkosti s poverhnostnymi volnami [Numerical simulation of fluid flows with surface waves. Novosibirsk: FUE Publishing House SB RAS; 2001: 394. (In Russ.)
[59] Dao, M.H., Tkalich, P. Tsunami propagation modelling — a sensitivity study. Natural Hazards and Earth System Science. 2007; (7):741–754.
[60] Horrillo, J., Kowalik, Z., Shigihara, Y. Wave dispersion study in the Indian Ocean-tsunami of December 26, 2004. Marine Geodesy. 2006; (29):149–166.
[61] Khakimzyanov, G.S., Shokina, N.Yu. Adaptive grid method for one-dimensional shallow water equations. Computational Technologies. 2013; 18(3):54–79. (In Russ.)
[62] Shokina, N.Yu., Khakimzyanov, G.S. An improved adaptive grid method for onedimensional shallow water equations. Computational Technologies. 2015; 20(4):83–106. (In Russ.)
[63] Shokina, N.Yu. To the problem of construction of difference schemes on movable grids. Russian Journal of Numerical Analysis and Mathematical Modelling. 2012; 27(6):603–626.
[64] Lovholt, F., Pedersen, G. Instabilities of Boussinesq models in non-uniform depth. International Journal for Numerical Methods in Fluids. 2009; (61):606–637.
Bibliography link:
Fedotova Z.I., Khakimzyanov G.S., Gusev O.I. History of the development and analysis of numerical methods for solving nonlinear dispersive equations of hydrodynamics. I. One-dimensional models problems // Computational technologies. 2015. V. 20. ¹ 5. P. 120-156
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