Yi-feng ZHANG*, Rui-jie LI

1. School of Civil Engineering, Tianjin University, Tianjin 300072, P. R. China

2. Tianjin Research Institute for Water Transport Engineering of Ministry of Transport, Tianjin 300456, P. R. China

3. Key Laboratory of Coastal Disaster and Defence of Ministry of Education, Hohai University, Nanjing 210098, P. R. China

Numerical solutions for two nonlinear wave equations

Yi-feng ZHANG*1,2, Rui-jie LI3

1. School of Civil Engineering, Tianjin University, Tianjin 300072, P. R. China

2. Tianjin Research Institute for Water Transport Engineering of Ministry of Transport, Tianjin 300456, P. R. China

3. Key Laboratory of Coastal Disaster and Defence of Ministry of Education, Hohai University, Nanjing 210098, P. R. China

The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schr?dinger (MNLS) equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave equations.

nonlinear water wave equation; parabolic mild-slope equation; spatial MNLS equation; numerical method

1 Introduction

Equations for water wave propagation in natural situations always present strong nonlinear features, making their numerical solutions considerably complicated and time-consuming.

The nonlinear parabolic mild-slope equation was derived by Kirby and Dalrymple (1983) using a multiple-scale perturbation method. In their study, the connection between the derived linearized version of the nonlinear mild-slope equation and a previous one was investigated. The equation was solved using the Crank-Nicolson finite difference method in the complex domain. This method can be used to predict wave propagation under moderately varying topographic conditions. Lin et al. (1998) and Tang et al. (2011) studied the numerical method to solve the same nonlinear equation. The iterative process was employed to deal with the nonlinear terms. This treatment increases the computational difficulty in different processes. The modified nonlinear Schr?dinger (MNLS) equation was obtained by Dysthe (1979). Lo andMei (1987) proposed a split-step pseudo-spectral method to solve the MNLS equation. This method was later used by Trulsen and Stansberg (2001) for solving the spatial MNLS equation. By solving this equation under the periodic boundary condition, Canney and Carter (2007) studied the instability of wave trains; Onorato et al. (2001) and Zhang et al. (2007a, 2007b) simulated deep-water wave evolution. Their studies showed that the split-step pseudo-spectral method was capable of improving the numerical stability and accuracy without using the iterative process. However, the periodic boundary condition was required at the second computational step of the discrete Fourier transform (DFT), which limited the practical application of the above-mentioned method.

Various studies have been carried out by researchers to improve numerical methods for solving nonlinear wave equations. This paper aims to modify the split-step pseudo-spectral method by eliminating the DFT process. The periodic boundary condition is no longer required, and the new method is expected to be easier to implement. To test its ability, the modified method was applied to solving the nonlinear parabolic mild-slope equation and spatial MNLS equation. The modified nonlinear dispersion relationship proposed by Li et al. (2003) was further used to process the solution to the nonlinear parabolic mild-slope equation. Wave propagations over elliptical topography (Berkhoff et al. 1982) and submerged circular shoal topography were simulated. Additionally, Keller’s wave flume experiment and the envelope solitary wave propagation that are governed by the spatial MNLS equation were also simulated.

2 Modification of split-step pseudo-spectral method

The split-step pseudo-spectral method proposed by Lo and Mei (1987) was used to solve the MNLS equation. The MNLS equation at each spatial step can be written by summing the linear and nonlinear terms:

where D is a complex function, x is the horizontal coordinate, and L and N are the linear and nonlinear operators, respectively.

Eq. (1) can be split into a linear equation and a nonlinear equation. At each spatial step, both equations are solved independently. The solution of the previous step is used as the initial condition for the next step. The first step is to solve the nonlinear terms:

whereΔx is the step size in the x direction. The second step is to apply the above solution to the linear terms using DFT, and then the complex function D(x +Δx ) can be obtained:

where Fand F?1are DFT and inverse DFT, respectively;; and P is determined by F(L (D ))= PF(D).

As the DFT function must be periodic, the periodic boundary condition should be satisfied in water wave simulation. However, it is difficult to adjust boundaries to be periodic ones in some cases. Therefore, it is impossible to simulate the water wave using the pseudo-spectral method with split steps under such conditions. To eliminate this limitation, the second step was modified by the finite difference method:

In this modified step, the calculation is not limited by the periodic boundary. Using the nonlinear term, the condition of Δx ＜ Ο(( Δ y)2) should be satisfied to keep the numerical method stable, where Ο ( ( Δ y )2) is the second-order infinitesimal function of the step size in the y direction.

3 Nonlinear parabolic mild-slope equation

The extended nonlinear parabolic mild-slope equation derived by Kirby and Dalrymple (1983) using a multiple-scale perturbation method is expressed as

where A is the complex harmonic amplitude of wave surface elevation; k is the wave number; c is the wave velocity, and c kω= , where ω is the angular frequency;gc is the velocity of wave groups, andgc kω=? ?;0kis the carrier wave number; andand h is the water depth.

Following the modified nonlinear dispersion relation proposed by Li et al. (2003), the angular frequency is expressed as

where p = tanh (k h); q =kh sinh (kh); ε is the wave steepness, and ε= kA; and g is the gravitational acceleration.

3.1 Numerical implementation

3.1.1 Difference scheme

Eq. (6) can be written in the following linear and nonlinear forms:

Using the central difference scheme to discretize the nonlinear terms, Eq. (8) becomes

where m (1 ≤ m ≤ M) and j (1 ≤ j ≤ J) are the grid serial numbers in the x and y directions, respectively, and M and J are the number of grid points in the x and y directions. The linear terms are discretized using the forward difference scheme:

3.1.2 Initial conditions

The incoming wave boundary condition at 0x= is

where0A is the initial amplitude of water waves.

3.1.3 Boundary conditions

At the lateral boundary for y= 0 and y = JΔy , the total reflection boundary condition is?A ?y= 0, which approximately follows (Kirby and Dalrymple 1983):

3.2 Numerical results

3.2.1 Simulation of elliptical topography

The elliptical topography and the cross-sections used in the model test conducted by Berkhoff et al. (1982) are shown in Fig. 1, where the data are the water depth and the serial number of the experimental cross-sections. The topography consisted of an elliptic shoal on a sloping-plane bottom with a slope of 1:50. The plane slope rose from a region at a constant depth of 0.45 m, and the entire slope was turned at an angle of 20o to a straight wave paddle. In the model test, the incoming wave propagated along the x direction, and its amplitude0A was 0.023 2 m. Hsu et al. (2008) and Zhao et al. (2009) simulated the wave propagation over this topography using other wave equations. In this study, the wave propagation was simulated with the model of the nonlinear parabolic mild-slope equation, described above. Fig. 2 showsthat the numerical results are in good agreement with the experimental data.

Fig. 1Topography and experimental cross-sections

Fig. 2Verification of results at cross-sections 1 through 8

3.2.2 Simulation of submerged circular shoal topography

The case of a submerged circular shoal on the bottom, which was first studied by Radder (1979), is used herein to test the applicability of the present method to solving the nonlinear parabolic mild-slope equation. Case 1 in Radder (1979) is used, with

where R is the shoal radius,0L is the incident wavelength,0H is the water depth outside the circular shoal, and1H is the water depth at the center of the circular shoal. The water depth h is given by

where r2=(x ?14)2+(y?10)2, and e0= (H0? H1) R2. The topography and cross-section layout are shown in Fig. 3. The incoming wave is along the x direction.

Fig. 3Topography and experimental cross-sections

The comparison of the present results and those of Kirby and Dalrymple (1983) at the central section is shown in Fig. 4(a). The calculated results agree with Kirby and Dalrymple’s in the range in front of and near the circular shoal. After passing through the shoal, the values are under-predicted by the present method, but the tendencies of the two results are consistent. Figs. 4(b) through 4(e) show that the calculated results agree with those of Kirby and Dalrymple at other cross sections.

Fig. 4Comparisons of calculated results and Kirby and Dalrymple’s results at sections 1 through 5

4 Spatial MNLS equation

The spatial MNLS equation derived by Dysthe (1979) using the perturbation method is expressed as

where t is time,φ is the averaged velocity potential, B is the complex amplitude of water waves, and B*is the complex conjugation of B.

4.1 Numerical implementation

Eq. (17) can be split into linear and nonlinear equations:

The central difference scheme is used to discretize the nonlinear term:

where m and n are the grid serial numbers in the x and y directions, respectively.

Using the forward difference scheme, the linear term can be transformed into

4.2 Numerical result

4.2.1 Simulation of Keller’s wave flume experiment

Keller (1982) experimented with periodic wave groups. The periodic wave groups were combined with two component waves. In the experiment, eight measuring points were arranged along the water trough. The frequencies of the two periodic waves were 1.406 Hz and 1.563 Hz, respectively, and the frequency of the carrier wave was set as 1.485 Hz, which was the average value of those two periodic waves. The corresponding wave number was k = 8.865 m?1. The amplitude of two periodic waves was 0.013 m, which was half of that of the carrier wave. The calculated water surface elevations η varying with time at different measuring points were compared with the experimental data, as shown in Fig. 5, where x is the distance from the measuring point to the wave maker. The horizontal axis shows a non-dimensional value of tεω π. It is clear that the numerical results are in good agreement with the experimental data.

4.2.2 Simulation of envelope solitary wave propagation

In order to study the invariable feature of the solitary wave in deep water, the envelope-soliton solution (Scott et al. 1973) was taken as the initial wave packet condition. It is expressed as follows:

In this calculation, the wave number k of the carrier wave is 4 m–1, the scale factor λ is 0.25, and the wave steepness ε is 0.1. Fig. 6 shows the calculated results of the propagation of solitary waves with different values of kx, which are 0, 10, 20, 30, 40, and 50. It is shown that the waveform almost remains constant. The results confirm that the waveform of solitary waves propagating in deep water is invariable, and the present method reproduces this feature.

Fig. 5Comparisons of calculated results and experimental data of water surface elevation at different measuring points

Fig. 6Wave profiles of envelope solitary waves

5 Conclusions

The split-step pseudo-spectral method was modified at its second step by avoiding transforming the wave height function into a frequency domain function. Using the modified method, the periodic boundary condition is no longer necessary. The nonlinear wavepropagations over Berkhoff’s elliptical topography, over the submerged circular shoal topography, and in Keller wave flume, as well as the propagation of envelope solitary waves were simulated with the modified method. The numerical results are in agreement with the experimental data. Therefore, the numerical method can be used to solve the nonlinear wave problem in nearshore and deep-water areas.

Overall, the numerical method of nonlinear water wave equations is feasible. With this method, the nonlinear terms can be solved without adopting iterative processes, and the method is proved to be very useful and easy to implement. It could be used in other nonlinear wave equations in the future.

Berkhoff, J. C. W., Booij, N., and Radder, A. C. 1982. Verification of numerical wave propagation models for simple harmonic linear water waves. Coastal Engineering, 6(3), 255-279. [doi:10.1016/0378-3839(82) 90022-9]

Canney, N. E., and Carter, J. D. 2007. Stability of plane waves on deep water with dissipation. Mathematics and Computers in Simulation, 74(2-3), 159-167. [doi:10.1016/j.matcom.2006.10.010]

Dysthe, K. B. 1979. Note on a modification to the nonlinear Schr?edinger equations for application to deep water waves. Proceedings of the Royal Society of London, Series A: Mathematical Physical and Engineering Sciences, 369(1736), 105-114. [doi:10.1098/rspa.1979.0154]

Hsu, T. W., Chang, J. Y., Lan, Y. J., Lai, J. W., and Ou, S. H. 2008. A parabolic equation for wave propagation over porous structures. Coastal Engineering, 55(12), 1148-1158. [doi:10.1016/j.coastaleng.2008.05.004]

Keller, J. B. 1982. Experiments on Nonlinear Wave Interaction. California: Stanford University.

Kirby, J. T., and Dalrymple, R. A. 1983. A parabolic equation for the combined refraction-diffraction of Stokes waves by mildly varying topography. Journal of Fluid Mechanics, 136(1), 453-466. [doi:10.1017/ S0022112083002232]

Li, R. J., Yan, Y. X., and Cao, H. S. 2003. Nonlinear dispersion relation in wave transformation. China Ocean Engineering, 17(1), 117-122.

Lin, G., Qiu, D. H., and Zou, Z. L. 1998. Numerical simulation of parabolic mild-slope equations. Journal of Dalian University of Technology, 38(3), 328-331. (in Chinese)

Lo, E., and Mei, C. C. 1987. Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study. Wave Motion, 9(3), 245-259. [doi:10.1016/0165-2125(87)90014-X]

Onorato, M., Osborne, A. R. Serio, M., and Bertone, S. 2001. Freak waves in random oceanic sea states. Physical Review Letters, 86(25), 5831-5834. [doi:10.1103/PhysRevLett.86.5831]

Radder, A. C. 1979. On the parabolic equation method for water-wave propagation. Journal of Fluid Mechanics, 95, 159-176. [doi:10.1017/S0022112079001397]

Scott, A. C., Chu, F. Y. F., and Mclaughlin, D. W. 1973. The soliton: A new concept in applied science. Proceedings of the Institute of Electrical and Electronics Engineers (IEEE) International Conference, 61(10), 1443-1483. [doi:10.1109/PROC.1973.9296]

Tang, J., Shen, Y. M., and Cui, L. 2011. Modeling coastal water wave propagation in vegetation field based on parabolic mild slope equation. Acta Oceanologica Sinica, 33(1), 8-11. (in Chinese)

Trulsen, K., and Stansberg, C. T. 2001. Spatial evolution of water surface waves: Numerical simulation and experiment of bichromatic waves. Proceedings of the Eleventh (2001) International Offshore and Polar Engineering Conference. Stavanger: The International Society of Offshore and Polar Engineers.

Zhang, Y. Q., Zhang, N. C., and Hu, J. P. 2007a. Numerical simulation and mechanism analysis of freak waves. Acta Oceanologica Sinica, 26(5), 16-124.

Zhang, Y. Q., Zhang, N. C., and Pei, Y. G. 2007b. Numerical simulation of freak waves based on the four-order nonlinear Schr?dinger equation. China Ocean Engineering, 21(2), 207-214.

Zhao, H. J., Song, Z. Y., Xu, F. M., and Li, R. J. 2009. A time-dependent numerical model of the extended mild-slope equation. Journal of Hydrodynamics, Ser. A, 24(4), 503-511. (in Chinese) (Edited by Ye SHI)

This work was supported by the Central Public-Interest Scientific Institution Basal Research Fund of China (Grant No. TKS100108).

*Corresponding author (e-mail: haizhongniao@163.com)

Received Feb. 14, 2012; accepted Sep. 6, 2012