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Numerical Study of Nonlinear Waves

  • Benyu Guo
Chapter

Abstract

Since Zabusky and Kruskal discovered the important behaviour of solitons by computation in 1965, numerical study of nonlinear waves has developed rapidly, and become one of the active branches of numerical analysis. There exist three main numerical methods. The first is the finite difference method. Zabusky, Kruskal [1] used the Leap-Frog scheme with the second order accuracy for the Korteweg- de Vries equation. Latter Vliegenthart [2], Greig, Morris [3] and Kuo Pen-yu [4] proposed dissipative, Hopscotch and conservative schemes respectively. Strauss, Vazquez [5] and Perring, Skyrme [6] applied the finite difference method to the Klein-Gordon equation and the sine-Gordon equation. The next is the finite element method. Wahlbin [7] developed the dissipative finite element scheme for the Korteweg-de Vries equation. Sanz-Serna, Christie [8] adopted the PetrovGalerkin approach, while Mitchell, Schoombie [9] constructed other schemes by using shift functions. The third is the spectral method. Gazdag [l0], Tappert [11], Canosa, Gazdag [12] Schamel, Elsässer [13], Watanabe, Ohishi, Tanaca [14], Abe, Inoue [15] and Guo Ben-yu [16] provided various spectral schemes for the Korteweg-de Vries equation. Recently Ma He-ping, Guo Ben-yu [17] proposed a new pseudospectral method with restraint operator. On the other hand, numerical experiment has played an important role in studying nonlinear wave equations. Bishop, Krumhansland, Trullinger [18] solved initial-boundary value problems of the sine-Gordon equation numerically. Chu, Xiang, Baransky [19] and Guo Ben-yu, Weideman [20] exhibited independently the solitary waves of the Korteweg-de Vries equation induced by boundary motion. Besides Kaup, Hansen [21] and Guo Ben-yu, Yan Xiao-pu[22] found some phenomena for the Schrödinger equation and the sine-Gordon equation by numerical investigations.

Keywords

Solitary Wave Nonlinear Wave Truncation Error Finite Difference Scheme Nonlinear Wave Equation 
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© Springer-Verlag Berlin Heidelberg 1995

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  • Benyu Guo

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