The nonlinear parabolic mild-slope equation is a useful method to study the nearshore wave problem. The equation with the dispersive relation that Li had modified was numerically simulated over the complex field. The numerical method doesn???t need transform the function to spectral-domain. This is different with that Lo used to simulate the MNLS equation with the nonlinear items. We simulated the surface water wave under the Berkhoff elliptic topography to verify the validity of the model, and the solution coincide quite well with the experimental data. The model is effective to simulate nonlinear water wave. The wave problem in Dachen island is studied with this model, and the distribution of wave height is predicted when the breakwater have been constructed. This method to solve the nonlinear items is very useful and easy to implement. The iteration and the limit of the periodic boundary are no longer necessary when solve the nonlinear parabolic mild-slope equation

Zhang, C.F.

Zhang, Y.F.

Hasanuddin University Repository

 767 
 
 
Proceedings of the 7th International Conference on Asian and Pacific Coasts  
(APAC 2013) Bali, Indonesia, September 24-26, 2013 
 
 
 
NUMERICAL SIMULATION OF NONLINEAR NEARSHORE WATER WAVE 
 
Y-F Zhang 1, C-F Zhang 2  
 
 
ABSTRACT: The nonlinear parabolic mild-slope equation is a useful method to study the nearshore wave problem. The 
equation with the dispersive relation that Li had modified was numerically simulated over the complex field. The 
numerical method doesn’t need transform the function to spectral-domain. This is different with that Lo used to 
simulate the MNLS equation with the nonlinear items. We simulated the surface water wave under the Berkhoff elliptic 
topography to verify the validity of the model, and the solution coincide quite well with the experimental data. The 
model is effective to simulate nonlinear water wave. The wave problem in Dachen island is studied with this model, and 
the distribution of wave height is predicted when the breakwater have been constructed. This method to solve the 
nonlinear items is very useful and easy to implement. The iteration and the limit of the periodic boundary are no longer 
necessary when solve the nonlinear parabolic mild-slope equation. 
 
Keywords: Nonlinear; parabolic mild-slope equation; dispersion relation; water wave; numerical simulation, breakwater.  
 
 
                                                 
1 Tianjin Research Institute for WaterTransport Engineering, Tianjin 300456, CHINA 
2 Shijiazhuang University, Shijiazhuan 050035, CHINA 
INTRODUCTION  
The nonlinear parabolic mild-slope equation was 
derived by Kirby (1983) using a multiple-scale 
perturbation method, and the connection between the 
linearized version and a previously derived 
approximation of the linear mild slope equation is 
investigated. The equation is solved using the Crank-
Nicolson difference method in the complex field. It can 
be used to predict the wave propagation on mildly 
varying topography. Wang (2006) improved a numerical 
model with this equation to study the distribution of 
wave direction. The nonlinear equation is linearized with 
some assumption when it is solved. The equation was 
divided to the real and imaginary parts by Lin (1998). 
Through, this avoids the difficulty of the calculation in 
the complex field, but there are two similar forms of 
nonlinear partial differential equations to solve. Due to 
the nonlinear terms of the equation, it always needs 
iteration to solve the equation numerically. This 
increases the the difficulty and computation in the 
discrete process. 
Lo [3] proposed a split-step, pseudo-spectral method 
to solve the modified nonlinear Schrödinger equation 
(MNLS), which was further implemented by Karsten 
(1987) for the spatial MNLS equation. This method had 
been shown to effectively improve the numerical 
stability and accuracy without the employment of 
iterative procedure. However, the periodic boundary 
condition must be needed in the second computational 
step of the Discrete Fourier Transform (hereafter DFT), 
largely limiting its applications within wider range of 
practical conditions. 
Dispersion relation plays an important role in the 
wave numerical calculation. Many scholars study the 
nonlinear dispersion relation, Li (2002, 2003) obtained a 
new formula of nonlinear dispersion relation, and 
compared with the dispersion relation obtained by Kirby. 
The accuracy of the nonlinear wave simulation is better 
with this nonlinear dispersion relation. Zhao (2009) 
based on hyperbolic mild slope equations and nonlinear 
dispersion relations to simulate the wave propagation. 
In this paper, the numerical method mentioned above 
is improved without DFT process. Therefore the periodic 
boundary condition is no longer necessary and the new 
method is expected to be relatively more convenient and 
feasible. This improved method is then applied to solve 
the nonlinear parabolic mild-slope equation. And the 
modified nonlinear dispersion relation of Li is used in 
this nonlinear wave model. The simulating results of this 
model were compared with experimental data sets for 
wave propagation over Berkhoff’s (1982) elliptical 
topography. The comparison between the numerical 
results and the experimental data reveals the validity of 
the model. Finally, the wave problem in Dachen island is 
studied with this model, and the distribution of wave 
height is predicted when the breakwater have been 
constructed. 
 
 
 
 
Y-F Zhang and C-F Zhang 
768 
 
THE NONLINEAR PARABOLIC MILD-SLOPE 
EQUATION 
The extended nonlinear parabolic mild-slope 
equation was derived by Kirby using a multiple-scale 
perturbation method. 
0||')()(
)())((22
2
0


AAKcckAcc
AkcciAcckkkAikcc
gyyg
xggxg
    (1) 
D
c
c
kK
g
)(' 3                                                     (2) 
kh
khkh
D
4
2
sinh8
tanh284cosh 
                        (3) 
k
c

 ，  
k
cg




                                            (4) 
Where x, y are horizontal coordinates, A  is complex 
harmonic amplitude of wave surface elevation,  is 
angular frequency, c is wave velocity, k is wave number 
and gc  is wave group velocity, A  is the module of A . 
Use the modified nonlinear dispersion relation by Li,  
)tanh()1( 22  qkhpgk                     (5) 
)tanh(khp                                                        (6) 
2)
)sinh(
(
kh
kh
q                                                   (7) 
 
NUMERICAL IMPLEMENTATION 
 
Numerical method and Difference scheme 
The split-step, pseudo-spectral method was used to 
simulate the MNLS equations by Lo.   
Every spatial evolution equation can be written as the 
sum of its linear and nonlinear terms: 
)()( ANALAx                                              (8) 
Where A  is a complex function, x is horizontal 
coordinate,
x
A
Ax


 , (.)L  and (.)N  are the linear 
and the nonlinear operators. This equation can be split 
into two equations: 
)(
'
ANAx                                                           (9) 
)(
''
ALAx                                                          (10) 
At each spatial step, both equations are solved 
independently, using the solution of the previous one as 
the initial condition for the next one. The first step is to 
solve the linear terms Eq. (9): 
))((
2
1
)()
2
1
(
~
xAxNxAxxA             (11) 
))
2
1
(
~
()()(
~
xxAxNxAxxA       (12) 
The second step is to take the above solution to the 
linear terms by using the DFT. 
)))(
~
(()( 1 xxAFeFxxA xiP            (13) 
Where 
1, FF are the DFT and the inverse DFT, and 
P  is determined by )())(( APFALF  . 
As the function of the DFT must be periodic, the 
periodic boundary is limited in the simulation of surface 
water wave. But it is difficult to adjust the boundary to 
be periodic. So sometimes it can not simulate the water 
wave by the split-step, pseudo-spectral method. The 
second step can be modified by the finite-difference 
method as: 
))(
~
(')()( xxAxLxAxxA           (14) 
Where 'L  is the transformation of the linear term L  
by the finite-difference. In this modified step there is no 
limit of the periodic boundary. With the nonlinear term, 
the step should be stratify 
2)( yx   to keep the 
method is numerically stable. It can simulate the 
nonlinear water wave equations generally. The modified 
method is applied to solve the nonlinear parabolic mild-
slope equation. 
Eq. (1) can be written into linear and nonlinear 
equations, 
0||'2 2  AAKiAx                                (15) 
0)()(
))((22 0


yygxg
gxg
AccAkcci
AcckkkAikcc
                  (16) 
Use the central difference scheme to discrete the 
nonlinear terms, 
jmjmjmjm
jm
AAxK
i
AA ,
2
,,,
,
2
1 ||'
4
~~


      (17) 
     
jmjm
jmjmjm AAxK
i
AA
,
2
1
2
,
2
1,,,1
~
|
~
|'
2
~~

  (18) 
Where m, j are the grid number in x and y coordinate. 
And forward difference scheme is used for linear 
terms, 
0
)(
2
)(
)()(
)()(
))((2)(2
2
1,,1,
,
,1,,1,
,
,,1
,
,0
,,1
,



















y
AAA
cc
y
AA
y
cccc
A
x
kcckcc
iA
cckkk
x
AA
kcci
jmjmjm
jmg
jmjmjmgjmg
jm
jmgjmg
jm
jmg
jmjm
jmg
    
(19) 
Substituting 1,,1, ,,  jmjmjm AAA  with 
1,1,11,1
~
,
~
,
~
 jmjmjm AAA respectively, and then 
jmA ,1 can be obtained. 
 
 
Numerical Simulation of Nonlinear Nearshore Water Wave 
 
 
769 
 
Initial conditions 
The incoming wave boundary condition 
( 0x ) is, 
sin
0,1
yikj
j eAA
    )1( Nj             (20) 
Where, 0A is initial amplitude of water wave, 
  is the wave direction to x direction, N is grid 
points number in y direction. 
 
Boundary conditions 
At the lateral boundary ( yNy  ,0 ), the total 
reflection boundary condition is 0/  yA . 
Here this boundary condition is approximated 
as that Kirby had done. 
1,,2,1, ,  NmNmmm AAAA )1( Mm   (21) 
Where, M is grid point number in x direction. 
 
Verification results 
The calculation topography and the experimental 
cross-section used in Berkhoff  verification test are 
shown in Fig.1. In the verification test, incoming wave 
direction along x and the amplitude 0
A
is 0.0232m. Fig.2 
shows that the numerical results are in fair agreement 
with the experimental data. This shows that the model is 
available to simulate the wave propagation. 
 
Fig.1 The graph of calculation topography and 
experimental cross-section 
 
 
Fig.2 Verification of section 1-8，*，the testing data; --
-----, calculating data 
THE WAVE SIMULATION OF DACHEN ISLAND 
 
Regional situation 
Dachen island is located in Chinese Taizhou bay and 
55km away from the Jiaojiang Haimen port. There are 
two islands: Shan Dachen island and Xia Dachen island. 
The Shan Dachen island area is 74.89km2. It’s 
surrounded by mountains, and the west side faces sea. 
There will construct a harbour of refuge and build a 
breakwater at the entrance of the port. Fig.2 shows the 
location of the project. Fig.3 shows the calculation 
region. 
 
 
Fig.3 Project location 
 
 
Fig.4 Calculation region 
 
Simulation result 
The grid step is 10m in the horizontal direction. We 
consider two incoming wave conditions that are 1:50 
year wave events according to the following 
specifications: 
 
 
Y-F Zhang and C-F Zhang 
770 
 
Table.1 The parameters of the incoming wave 
tes
t 
wave height 
(m) 
wave 
period(s) 
wave 
direction(degree) 
1 2.68 9 90 
2 2.68 9 60 
 
Fig.5~Fig.8 show the wave height distribution. The 
wave height is above 1.5m at somewhere of the port in 
test 1 without breakwater. After the breakwater has been 
built, the wave height is smaller as the covering of the 
breakwater. It’s about 0.5m. In test 2, as the impact of 
coastline and the wave direction, the wave height is 
smaller and about 0.5m. And the breakwater plays a 
small role in covering the incoming wave of test 2. 
  
Fig.5 Wave height distribution without breakwater (test 
1) 
 
Fig.6 Wave height distribution with breakwater (test 1) 
      
Fig.7 Wave height distribution without breakwater (test  
2) 
    
Fig.8 Wave height distribution with breakwater (test 2) 
 
CONCLUSIONS 
The nonlinear parabolic mild-slope equation is 
numerically solved by the modified split-step, pseudo-
spectral method. As the modification in the second step, 
the improved method doesn’t need transform the 
function to spectral-domain. The periodic boundary 
condition is no longer necessary and the new method is 
more convenient to solve the nonlinear wave equations. 
And the modified dispersion relation is used with the 
nonlinear parabolic mild-slope equation. 
The surface water wave under the Berkhoff elliptic 
topography is simulated to verify the validity of the 
model, and the solution coincide quite well with the 
experimental data.The wave problem in Dachen island is 
studied with this model, and the distribution of wave 
height is predicted when the breakwater have been 
constructed. The breakwater can decrease the wave 
height in the port when the wave direction is along the 
bay. 
With this method the nonlinear terms can be solved 
without iterating, and it is very useful and easy to 
implement.It also should be used in other nonlinear 
water wave equations in future. 
 
REFERENCES 
Kirby J T, and Dalrymple R A. (1983). A parabolic 
equation for the combined refraction-diffraction of 
Stokes waves by mildly varying topography. J Fluid 
Mech, 136, 453-466. 
Wang Hong-chuan, Zuo Qi-hua, Pan Junning.(2006) . 
Numerical analysis of wave direction on the 
nonlinear parabolic mild slope equation. Journal of 
Hydrodynamics ,Ser A, 21(1):139-144. 
Lin Gang, Qiu Da-hong, and Zou Zhi-li. (1998). 
Numerical simulation of parabolic mild-slope 
equations. Journal of Dalian University of 
Technology, 38(3): 328-331 
 
 
 
Numerical Simulation of Nonlinear Nearshore Water Wave 
 
 
771 
 
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. 
Li Rui-jie, Wei Shou-lin, Yu Feng-xiang. (2002). Wave 
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Nonlinear dispersion relation in wave transformation. 
China Ocean Engineering, 17(1), 117-122. 
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(2009). A time-dependent numerical model of the 
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Berkhoff J C W, Booij N, and Radder A C. (1982). 
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Engineering, 5,255-279. 
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Numerical simulation of nonlinear nearshore water wave

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Numerical simulation of nonlinear nearshore water wave

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