kdv and fkdv model for the run-up of tsunamis via lattice...

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 14338-14347

© Research India Publications. http://www.ripublication.com

14338

KDV and FKDV Model for the Run-up of Tsunamis via Lattice Boltzmann

Method

Sara Zergania,b, J. H. Leec, Z. A. Aziza,b and K. K. Viswanathan a,d*,

aUTM Centre for Industrial and Applied Mathematics, Ibnu Sina Institute for Scientific & Industrial Research, Universiti

Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia. bDepartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia. cDepartment of Naval Architecture and Ocean Engineering, Inha University, 100 Inharo. Nam-gu, Incheon 22212, South Korea.

dKuwait College of Science and Technology, Doha District, Block 4, P.O. Box No. 24275, Safat 13133, Kuwait.

*Corresponding author Orcid: 0000-0003-4470-4774

Abstract

An efficient implementation of the Lattice Boltzmann method

(LBM) for the numerical simulation of the run-up tsunamis,

based on Korteweg-de-Vries equation (KdV) and forced

Korteweg-de-Vries equation (fKdV) are presented. Numerical

results are found to be in good agreement with the theory.

Keywords: Lattice Boltzmann, KdV and fKdV equations,

run-up of tsunami, numerical method

INTRODUCTION

In the fields of physics and mathematics, the significant role

that nonlinear partial differential equation (NPDE) plays

cannot be over emphasized, since their nonlinear behaviors

contain various fascinating and valuable hidden characteristics

of physical systems. The availability of the accurate solutions

has the capability of enhancing our knowledge of the NPDEs

model that is processed dynamically, and as well as the

mechanism for the intricate physical phenomena.

Consequently, it becomes an imperative feature to probe and

create accurate solutions for NPDEs. However, owing to the

fact that the accurate solutions barely occur in these stern

situations, a large number of the NPDE investigations are

conducted using suitable numerical techniques devised

purposely for nonlinear issues.

As such, for about 150 years, a nonlinear PDE of third order

known as the KdV equation has been of great significance to

the researchers. Investigations in the area of unusual water

waves occurring in the shallow and narrow waterways,

usually canals, employ the use of the KdV equation. John

Scott Russell in 1844 detected a phenomenon along the

Edinburgh-Glassdow canal, during the investigation on the

most effective strategy for canal boats. The result of the

investigation revealed that when water in the canal is put in

motion as a result of a boat being pulled by a pair of horses

amassed in a state of fierce actions and then trolled onward

with abnormal swiftness reshaped as a huge retiring altitude,

that is circumnavigated, smooth, and a distinct pile of water

that prolonged its passage along the canal without

transformation or dwindling speed. However, after one or two

miles, the altitude slowly faded away. John Scott Russell

identified this remarkable and stunning discovery as the Wave

of Translation. The KdV equation model, gives rise to the

dynamics of solitary waves. The KdV equation consists of

non-linear, distributive, non-dispelling equation containing

soliton solutions. In the modeling of tsunamis, solitary waves

are normally used particularly in the experimental and

mathematical research since the 1970s. The recurrent

assumption that solitary waves could be used in the modeling

of significant characteristics of potential tsunamis to the beach

and shorelines has been in existence since the early 1970s, as

well as the notion that since the theories originated from the

KdV equation, it has the capacity to describe the appropriate

input of waves for physical or mathematical models of

tsunamis.

Literature revealed that several researchers [1,2] have used

this equation. The recent NOAA (National Oceanic and

Atmospheric Administration product information catalogue)

Technical Memorandum is another illustration of the

popularity of this concept [3] that examined crucial analytical

and experimental yardsticks for numerical models for

tsunamis [3]. In this memorandum 16 out of 45 references

cover solitary waves, and although one do acknowledge that

these waves can be helpful for the verification of a numerical

model, the memorandum leaves one with the impression that

the solitary wave should be the preferred model of a tsunami.

However, in as much as we recognize the fact that these

waves have the capability of aiding the authentication of a

numerical model, the memorandum suggests that solitary

waves are the preferred model for tsunamis. The use of

solitary waves as the principal wave of tsunamis has long

existed. The theory of KdV equation is dispersive as well as

nonlinear; besides, the only waves with the distinctive

property of static shape are the solitary waves. Several

numerical approaches drawn from the NPDEs were devised in

the previous decades; which comprised the finite difference

approach, the heat balance integral approach, the finite

element approach, the spectral approach and the variational

iteration approach [4]. Distinct from the conventional

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numerical systems based on the discretizations of macroscopic

continuum equations, the Lattice Boltzmann method (LBM) is

a comparatively new method based on microscopic models

and mesoscopic kinetic equations. In the research of nonlinear

equations and complex systems evolution, the LBM has been

enormously successful, particularly in liquid mechanics [5]. It

is noteworthy to state that most of the benefits of molecular

dynamics are delivered by LBM comprising of clear physical

pictures, simplicity in integrating intricate boundary

conditions, and effortlessness of programming [6].

Likewise, owing to the equilibrium allocation, it is possible

for functions to be computed simultaneously, where the model

could be instinctively improved to match computing process.

Furthermore, its prospects are all encompassing, stretching

from turbulent flow to multi-phase flows, multi-component

flows, particle suspensions, quantum mechanics and

hemodynamics. The capabilities of stimulating the nonlinear

systems by the LBM was revealed recently, which entailed

convection-diffusion equation, reaction-diffusion equation,

Burgers’ equation, MKdV equation and KdV-Burgers’

equation (KdVB) [4, 6]. Conversely, the process of solving

NPDEs in most current LBMs posed problematic, involving

how higher accuracy could be achieved as well as further

intricate nonlinear terms in NPDEs. Findings reveal that

numerical results are exceptionally in agreement with the

analytical solutions [4, 6]. However, compared with the

computational fluid dynamics methods, the LBM is easy for

programming, basically comparable, as well as simple in

incorporating intricate boundary conditions. The forced

Korteweg-de Vries (fKdV) equation refers to the KdV

equation that has a forcing term and develops as a model for

numerous physical situations that includes tsunami run-up.

But the fKdV equation is a nonlinear evolution equation that

joins numerous effects that include forcing; ( )f x ,

nonlinearity; xUU , and dispersion; xxxU terms. The

conventional analytical approach, which includes inverse

scattering approach and Backlund transformation, has ceased

to function on forced system. Consequently, the method

identified for solving the fkdV equation, appears to be the

approximate and numerical solutions. Therefore, in this paper,

we solved the fKdV equation numerically using lattice

Boltzmann method (LBM). It is expected that the lattice

Boltzmann model could be used to search out some new

solutions for the fKdV equation.

METHODOLOGY

The KdV equation, given by

6 0t x xxxu uu u (1)

which explains the development of prolonged waves (with

large length and measurable amplitude) along a channel with a

rectangular cross section. Here u denotes the amount of

wave, and tu and xu are the partial derivatives with respect to

𝑡 and 𝑥, respectively. The quantity tu denotes the vertical

velocity of the wave at ( , )x t , xu explains the rate of change

in amplitude with respect to x , and xxxu is a dispersion term.

This means that if 𝑢 is the amplitude of a wave at some point

in space, and then xu is the slope of the wave at that point.

The existence of solitary waves occurs because of the

balancing effects of xuu and xxxu in Eq. (1). The nonlinear

term 6 xuu in Eq. (1) is important because the amplitude of the

wave depends on its own rate of change in space; it also

represents steepening. The term xxxu suggests dispersion of

different frequency components. Dispersive waves are usually

characterized by solutions

( )( , ) ikx iwtx t Ae (2)

for linear problems, where 𝑘 is the wave number, 𝑤 is the

frequency, and A is the amplitude. In fact, the dispersion

relation, written as ( )w w k , coupled together with the

nonlinear term mentioned previously, is what produces the

balance between nonlinearity and dispersion and generates

solitary waves (instead of the formation of other known

waves) [7]. Solitary wave is one of the most interesting wave

phenomena existing in the natural world. According to Eq.

(1), solitary water wave is described as “a wave that consists

of a single elevation (a rounded, smooth, and well-defined

heap of water), neither proceeded nor followed by another

elevation (or depression)”. This description is only in one

occurrence of solitary waves; but also occurs in different

physical mediums. Solitary waves are capable of possessing

high intricate dependence on both space and time, while

certain waves are capable of being modeled through simple

expressions relating the amplitude, frequency, and speed to

the medium through which the waves propagate.

Lattice Boltzmann Model of KdV Equation

The lattice Boltzmann models for this kind of equation must

be constructed respectively, if these are to be applied in the

simulation of a nonlinear evolution equation. In this paper, we

have retrieved the corresponding macroscopic equation by

using the single relaxation form of the lattice Boltzmann

equation, the Chapman–Enskog expansion, and the multi time

scale technique, i.e. by defining the macroscopic variables and

selecting the local equilibrium distribution functions. Besides,

the higher approximation numerical results of the lattice

Boltzmann method are obtainable through the selection of the

higher order torques of the local equilibrium distribution

functions appropriately, and bringing the truncation error

higher [6].

The macroscopic quantity ( , )u x t is defined by

( , ).u f x t (3)



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The conservational condition of u requires that

(0) ( )( , ) ( , ).equ f x t f x t (4)

The particle distribution function satisfies the lattice

Boltzmann equation

(0)1( , ) ( , ) ( , ) ( , )f x e t f x t f x t f x t (5)

where ( ) ( , )eqf x t the equilibrium distribution function is at

time t , position x and is the single relaxation time factor.

is the step of time and the Knudsen number, 1 . The

stability of the equation requires that 1

.2

The distribution

function ( , )f x t is the probability of finding a particle with

velocity e , here 0,1,...,4 . If the spatial step is set to k ,

the possible values of the velocity e , are ( 2 , ,0, ,2 )k k k k ,

where k is the scale factor of the time and the spatial step.

Applying Taylor expansion and Chapman-Enskog expansion

to the Eq. (5), and retaining terms up to, we get

0

22

33

445

1( , ) , ( , ) ( , )

!

, ( , ) , ( , )2

, ( , )6

, ( , ) ( ).24

n

n

f x e t e f x t f x tn t x

se f x t e f x t

t x t x

se f x t

t x

se f x t O

t x

(6)

Next, the Chapman-Enskog expansion is applied to f

under the assumption that the mean free path is of the same

order of . Expand f and (0)f

( )

0

(0) (1) 2 (2) 3 (3) 4 (4) 5

( , )

( )

n n

n

f x t f

f f f f f O

(7)

where, f is (0)f .

To discuss changes in different time scales, we introduce

, 0,1,2,3,4t t , thus

2 3 4

0 1 2 3 4, , , , ,t t t t t t t t t t (8)

t

can be written in the form

2 3 4 5

0 1 2 3 4

( ).Ot t t t t t

(9)

The equation to the order of is

(0) (1)1 .e f ft x

(10)

The equation to the order of 2 is

2

(0) (0)

1 0

(2)

11

2

1.

f e ft t x

f

(11)

The equation to the order 3 is

(0) (1)

2 1

3

2 (0) (3)

0

12

1 1.

6

f ft t

e f ft x

(12)

The equation to the order 4 is

2

(0) 2 (0)

3 0 1

(0)

2

4

3 2 (0)

0

(2) (4)

1

12 2

4

1 2

3 7 1

2 12 24

1 11 .

2

f e ft t x t

e ft t x

e ft x

f ft

(13)

Eqs. (10), (11), (12), (13) are so-called series of Lattice

Boltzmann equations in different time scales. Using the above

equations it can be modified as

2

(0) (0) (0)

0

3

2 2 (0)

0

2

3 2 (0)

1 0

4

3 2 (0)

0

1

2

1

6

12 2

4

3 7 1

2 12 24

f e f e ft x t x

e ft x

e ft t x

e ft x

4

4 1 ( )

1

1.j j

j

O f

(14)

We get ( ) 0 ( 1,2,3,4)jf j , thus the right-hand side

of the above Eq. (14) is equal to 0.

In Eqs. (10)–(14) we find four polynomials or (Bernoulli

polynomials) of the relaxation time factor.

1 1C (15) , 21

2C

(16)



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2

3

1

6C

(17)

3 2

4

3 7 1

2 12 24C

(18)

Eqs. (15)–(18) are the first six Bernoulli polynomials. The

expressions given above are in full agreement with results in

the literature [8]. They can be used to indicate coefficients of

the dispersion term and the dissipation term to the KdV

equation.

Equilibrium Distribution Function

Some moments of the equilibrium distribution function are

defined as the following and we select the torques of the local

equilibrium distribution functions as

( , ) ( , )u x t f x t (19)

(0) 21( , ) ( , )2

S x t f x t e au (20)

(1) 2 2 31( , ) ( , )3

S x t f x t e a u (21)

(2) 3 3 41( , ) ( , )4

S x t f x t e a u u (22)

(3) 4 4 5 21( , ) ( , ) 25

S x t f x t e a u a u (23)

The 5-velocity model encompasses the following velocities:

0 0,e 1 ,e c 2 ,e c 3 2 ,e c 4 2e c . The equilibrium

distribution function can be expressed in terms of the

moments of order 0, 1, 2, 3 and 4 by solving Eqs. (19)–(23),

(0) (0) 3 (1) 2 (2) (3)1 41

4 46

f S c S c S c Sc

(24)

(0) (0) 3 (1) 2 (2) (3)2 41

4 46

f S c S c S c Sc

(25)

(0) (2) (0) 3 (3) (2) 23 41

2 224

f S c S c S S cc

(26)

(0) (2) (0) 3 (3) (2) 24 41

2 224

f S c S c S S cc

(27)

(0) (0) (0) (0) (0)

0 1 2 3 4f u f f f f (28)

Here 23/ C is the parameter to be determined by

selecting . We have obtained a lattice Boltzmann model for

the KdV equation with the third-order truncation error by

using four equations in different time scales.

Recovery of the KdV Equation

Summing Eq. (14), we have the KdV equation is written as

32 3

3

1( )

2

u uau O

t x x

(0)32 3 3

3 23

2

3 4 (0) 4

4

12

2

( ).

fu uau C C

t x tx

C f O

Thus

43 4 (0) 4 2

4 4 4

4 4 (3)4 5 2

4 4

3

2

1 10.

5 2

C f C a ux

Sa u au

x x

The KdV equation with the fourth-order accuracy of

truncation error is given by

32 4

3

1( ).

2

u uau O

t x x

(29)

Lattice BGK Model

The proposition of the lattice Boltzmann scheme with an

amending function for the nonlinear fKdV equation is meant

for the extension of the lattice Boltzmann method for dealing

with more nonlinear equations. The nonlinear phenomena as

modeled by fKdV equation emerge in several aspects of the

scientific fields that include fluid dynamics and run-up

tsunami. The process of solving fKdV in various existing

lattice Boltzmann models, posed problematical. As such, we

created a higher order lattice Bhatnager-Gross-Krook (BGK)

model with an amending function as well as a source term.

The created model improved its accuracy (to the 5( )O

order), and made the problem more integrated and more easy

to solve. Utilizing the Taylor and Chapman-Enskog

expansions, the nonlinear fKdV equation has been accurately

retrieved from the lattice Boltzmann equation. An appropriate

selection of collision or equilibrium distribution ensures that

the lattice Boltzmann model is able to retrieve the fKdV of

interest, which is a method recently developed based on a

mesoscopic kinetic equation for the particle distribution

functions. In comparison with the predictable numerical

methods, the LBM offers a number of benefits, comprising

geometrical flexibility, clear physical pictures, ease in the

incorporation of intricate boundary conditions, ease of

programming and numerical efficiency, which signify the

effectiveness and flexibility of the current methodology that

ensure realistic application.

We consider fKdV as

( ).t x xxxu uu u F u (30)



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where and are real constants. The lattice Boltzmann

equation with an amending function and a source term is as

follows [9]:

(0)

2

1( , ) ( , ) ( , ) ( , )

( , ) ( )5

i i i i i

i

f x e t f x t f x t f x t

sh x t F u

(31)

where ( , )if x t and (0) ( , )if x t are defined as density

distribution function and equilibrium distribution function,

respectively, ( , )ih x t is an amending function, c is a constant,

and is the dimensional relaxation time. The stability of the

equation requires that 0.5 [9]. The macroscopic variable

u meets the following conservation laws.

(0)( , ) ( , ) ( , )).i ii iu x t f x t f x t (32)

Then, through choosing appropriate local equilibrium

distribution, we can retrieve the corresponding macroscopic

equation correctly by using the Chapman-Enskog expansion.

Indeed, applying Taylor expansion to the left hand of Eq. (31)

and retaining terms up to 5( )O , we get

0

22

3 43 45

(0) 2

( , )

1, ( , )

!

( , ) , ( , ) , ( , )2

, ( , ) , ( , ) ( )6 24

1( , ) (

i i

n

i i

n

i i i i i

i i i i

i i i

f x e t

e f x tn t x

sf x t e f x t e f x t

t x t x

s se f x t e f x t O

t x t x

f x t f h

, ) ( ).5

sx t F u

(33)

Using the Chapman-Enskog expansion (7)-(9) and

2

1( ) ( )F u F u (34)

where the Knudsen number is defined as L

, is the

mean free path, and L is the characteristic length, which can

be taken as the time step t , and ( ) ( 1,2,...)nif n are the

non-equilibrium distribution functions, which satisfy the

solvability conditions:

(0)( , ) 0 ( 1,2,...), .ii f x t n t (35)

Let and substituting Eq. (34) and Eqs. (7)-(9) into Eq. (33),

we have

2

(0) (0) (0)

0

3

2 2 (0)

0

2

3 2 (0)

1 0

3 2 (0)

0

1

2

1

6

12 2

4

3 7 1

2 12 24

i i i i ii i i

i ii

i ii

i ii

f e f e ft x t x

e ft x

e ft t x

e ft x

4

341 ( ) 2

1

( )

1( , ) ( ).

5

j j

i iij

O

sf h x t F u

(36)

Comparing the two sides of Eq. (36) and treating the term in

the order of gives the equations to the order of as

2

(0) (1)

0

1i i ie f f

t x

(0) (1) (1) (0)1

i i i i i ie f f f e fx x

(1) 2 (0) 2 (1) 3 (0)

3 (1) 4 (0) .

i i i i i i i i

i i i i

e f e f e f e fx x

e f e fx

(37)

The equations to the order of 2 are

2

(0) (0) (2)

1 0

2(0) (1) 2 (0) (2)

2

1

2(0) 2 (0) (2)

2

1

1 11

2

1 1

2

1 11

2

i i i i i

i i i i i i i

i i i i i

f e f f ht t x

f e f e f f ht x x

f e f f ht x

(39)

2

(2) (0) 2 2 (0)

2

1

11

2i i i i if h f e f

t x

2(2) (0) 2 3 (0)

2

1

22 (2) 2 2 (0) 2 4 (0)

2

1

11

2

11

2

i i i i i i i i

i i i i i i i i

e f e h e f e ft x

e f e h e f e ft x



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3

(0) (1) 2 (0)

2 1 0

(3) 1

2(0) (1) (2) 2 (1)

2

2 1

3(0) 3 (0) (3) 1

3

1

(0)

2 1

1 12

6

( )1

5

1

2

( )1 1

6 5

12

i i i i

i

i i i i i i

i i i i i

i

f f e ft t t x

F uf

f f e f e ft t x x

F ue f e f f

t x x

ft t

(1)

32 3 (0) (3) 1

3

(3) (0) (1)1

2 1

32 2 3 (0)

3

(3) (0) 2 (0)1

2 1

2 2

( )1 1

6 5

( )1 2

5

1

6

( )2 1

5

i i i

i i i

i i i

i i i i

i i i i i i i

i i

f e hx

F ue f f

x

F uf f f

t t

e h e fx x

F ue f e e f e f

t t x

e hx

32 4 (0)

3

1.

6i ie f

x

(39)


2

(0) 2 (0)

3 0 1

(0)

2 1

4

3 2 (0)

0

(2) (4)

1

12 2

4

1 2

3 7 1

2 12 24

1 11

2

i i i

i i

i i

i i

f e ft t x t

e ft t x t

e ft x

f ft

2(0) (2) (1) (3) (0)

2

3 1 2 1

2 3(0) 2 (2) 3 (1)

2 3

2 1

2 42 (0) 4 (0) (4)

2 4

1

1

2

1 1

2 6

1 1 1.

2 24

i i i i i i

i i i i i i

i i i i i

f f f e f ft t t x t

e f e f e ft x tx x

e f e f ft x x

(40)

Substituting Eqs. (37), (38), and (39) into Eq. (40), we obtain

(0) (2) (1) 1

3 1 2

2 22 2 (0) 2

2 2

1

( )1 11 2

2 5

1 12 2

4 2

i i i i

i i i i

F uf f f e

t t t x

e f e ht x x

(41)

4 3

3 2 4 (0) 3 (1)

4 3

(4)

1

3 7 1 1

2 12 24 6

1 1

2

i i i i

i i

e f e fx x

h ft

To

recover Eq. (30), we select the local equilibrium distribution

functions to satisfy

(0) ( , ) 0i ii f x t e (42)

(0) 2( , ) 0i ii f x t e (43)

(0) 3( , )i ii f x t e u (44)

(0) 4( , ) 0i ii f x t e (45)

where is some parameter to be determined.

Meanwhile, the amending functions ( , )ih x t ( 0,1,2,3,4)i

satisfy

0ii h (46)

2 1

1 2

n

i iie h u u (47)

2

i iie h u (48)

where 1 2, , and are some constants to be determined.

Summing up the two hands of Eqs. (37), (38) and (41) with

respect to i , and using Eqs. (32), (42)-(45) and (46)-(48), we

obtain

2

1

( ) : 0u

Ot

(49)

3 1 22

2

1

( ) : 2 1

1( )

6

n

x x

xxx

uO uu n u u

t

u F u

(50)

4

2

1( ) : 0.

2xx

uO u

t

(51)

Finally, 2 3(49) (50) (51) reads

2 2 2 21 2

2

12 1

6

10 ( )

2

n

t x x xxx

xx

u uu n u u u

u F u

(52)

(0) (0) (0)

1 2 3

(0) (0)

4 0

, , ,6 6 12

,12

f u f u f u

f f u

(53)

where is the only free parameter to be determined. The

numerical accuracy and stability are improved by choosing



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the proper parameter . From Eq. (46)-(48), the amending

functions can be defined. For simplicity, only one case for ih

is given:

2 2 1

0 1 1 2

2 1 2 2

2 1 2 3 4

1 1 16 , 4 ,

2 2 2

1 1 14 , , .

2 2 2

n

n

h u u h u u u

h u u u h u h u

(54)

To recover Eq. (30), just let

2

12c t (55)

2 21n c t (56)

2 31

2c t

(57)

3 2 21

6c t

(58)

Then

3 2

1 1

2 12 c t

(59)

2 3

.1

2c t

(60)

RESULTS AND DISCUSSION

In the following, to test LBM model proposed in the above

section, numerical simulations of KdV equation and fKdV

equation are performed.

Test 1: A test problem, two-solitons problem, with initial

function is

2 21 1 1 2 2 26

( , ) 3. .sech . 3. .sech . ;

t x xxxu uu u

u x t c k x d c k x d

The numerical results of the lattice Boltzmann model Figure 1

until Figure 5 are the numerical results at different times.

Parameters are: lattice size 200,M 1 0.3,c 2 0.1,c

1 2, 6,d d 1

,xM

0.005,x

tc

1, 0.001,

0.0000001v .

-100 -50 0 50 100-0.5

0

0.5

12D and 3D Kdv using LBM D1Q5

x cell

velo

city

--t == 1.00

-100 -500 50

100-1

0

1-0.5

0

0.5

1

Figure 1: The numerical results of the lattice Boltzmann

model KdV at time 1.00t

-100 -50 0 50 100-0.5

0

0.5

12D and 3D Kdv using LBM D1Q5

x cell

velo

city

--t == 2.00

-100 -500 50

1000

0.5

1-0.5

0

0.5

1


model KdV at time 2.00t

-100 -50 0 50 100-0.5

0

0.5

1

1.52D and 3D Kdv using LBM D1Q5

x cell

velo

city --t == 50.00

-100 -500 50

1000

50-1

0

1

2


model KdV at time 50t



14345

-100 -50 0 50 100-0.5

0

0.5

1


x cell

velo

city --t == 100.00

-100 -500 50

1000

50

100-1

0

1

2



-100 -50 0 50 100-0.5

0

0.5

1


x cell

velo

city --t == 150.00

-100 -500 50

1000

100

200-1

0

1

2



Test 2: In this case, we can obtain a special type wave

solution:

1 0

22

1

( )

.arctan( ) .

( , ) 12. . .sech

t x xxxu uu u F u

at b At b t b

u x t k k x d at

The numerical results of the lattice Boltzmann model Figure 6

until Figure 11 are the numerical results at different times.

Parameters are: lattice size 1 3.0,b 1.0,b 2

1 4. ,d k

, 1,t k 3,A 1, 0.001, 0.00001v .

-4-2

02

4

-4

-2

0

2

4

-1

0

1

2

3

--t == 1.00

x cell

3D fKdv using LBM

time

velo

city


model fKdV at time 1.00t

-4-2

02

4

-4

-2

0

2

4

-1

0

1

2

3

--t == 54.00

x cell

3D fKdv using LBM

time

velo

city


model fKdV at time 50t

-4-2

02

4

-4

-2

0

2

4

-1

0

1

2

3

4

--t == 207.00

x cell

3D fKdv using LBM

time

velo

city





14346

-4-2

02

4

-4

-2

0

2

4

-1

0

1

2

3

4

--t == 311.00

x cell

3D fKdv using LBM

time

velo

city



-4-2

02

4

-4

-2

0

2

4

-2

0

2

4

6

--t == 417.00

x cell

3D fKdv using LBM

time

velo

city



-4-2

02

4

-4

-2

0

2

4

-2

0

2

4

6

8

--t == 500.00

x cell

3D fKdv using LBM

time

velo

city



In this paper, the numerical solution to the KdV equation was

acquired, with a forcing term, using LBM method for the

purpose of ratifying inundation models. It can be observed

that the same type of solution can be excited by differently

forced term. The reference [10] illustrated different types of

forcing sources for KdV by using Hirota’s direct method for

tsunami and bottom landslides movements. One example has

been selected from Zhao and Guo [10], “The First Type One-

Soliton Solution" and applied by using LBM with an

amending function (see test case 2). In order to use these test

cases for LBM, thus for an accurate description of model, it is

necessary to detect adequate units, size and parameters. The

D2Q5 squared lattice is preferred in use for KdV and fKdV

and it is employed in these test cases. To achieve a proper

lattice solution, a lattice (100 100) is used. According to the

numerical codes, benchmarking variety of lattices is possible.

Difference in the results clearly exists but slightly. The

method employed is sensitive to choice of values for the

single relaxation time over the range considered. For single

relaxation time 1 , LBM will not be stable and generates

unphysical oscillations. For single relaxation time 1 ,

LBM will be stable and depends on value which shows

different quantities. The LBM may suffer instability like any

other numerical methods, but by using suitable time relaxation

values, lattice size and time step (iterations), such instabilities

could be minimized. Another option that is as well as

important for stability of LBM is the kinematic viscosity. The

kinematic viscosity must be negative 2 1

06

. In

comparing LBM numerical results (see LBM simulation table

below), with analytical solutions Zhao and Guo [10], two

results were randomly picked from the fKdV part, thereby,

giving an observed error of the maximum absolute error is

0.10344 and the global relative error is 0.015587, refer to

Table 1. The benefits of the lattice Boltzmann method for the

forced KdV equation are of greater flexibility in the choice of

different forcing sources. Its behaviour and velocity field are

shown in Figure 1-5 and Figure 6-11. Result at different

times, match eventually with Zhao and Guo [10]. The

proposed model’s effectiveness and precision is authenticated

using LBM numerical simulations, which established that the

LBGK results are in reasonable agreement with the analytical

solution.

The maximum absolute error (MAXE), which is defined as

max xMAXE U u , and the global relative error (GRE),

defined as xU u

GREu

where xU and u are the

numerical and analytical solutions, respectively. The next, two

results for the fKdV equations are listed as in Table 1 below.



14347

Table 1 Comparison of the numerical and analytical solution

u xU xU u

1 0.2504 0.250176 0.000224

2 0.2347 0.234481 0.000219

3 0.2192 0.219096 0.000104

4 0.2033 0.203222 7.77E-05

5 0.1867 0.186657 4.28E-05

6 0.1696 0.169579 2.13E-05

7 0.1524 0.152363 3.73E-05

8 0.1354 0.135444 4.37E-05

9 0.1192 0.119223 2.31E-05

10 0.104 0.104019 1.91E-05

max 0.10344xMAXE U u

0.015587xU u

GREu

Figure 1 until Figure 5 show the phenomenon of swallowing

and spitting in addition investigation how bigger soliton reach

and collide with smaller soliton in the process. The behaviour

and velocity of “The First Type One-Soliton Solution” are

shown in Figure 6 until Figure 11. One can see on these plots

oscillation and velocity grow up and significant change

happened after 200t . The changes on the shape and celerity

of the soliton after 200t are visible. Our numerical results

show that their velocity field of function u increases when

t .

CONCLUSION

The exclusive use of the Korteweg-de-Vries equation (KDV)

model in the evaluation of the run-up of tsunamis has been in

existence prior to the late 1990s. The fKdVB model was

recently motivated and it produces tsunami waves that caused

the initial receding of nearby shorelines before its

encroachment. According to Synaolakis on solitary wave

(fKdV) evolution and run-up, it is conceivable based on linear

theory to originate precise results for the evolution and run-up

of solitary waves. Tsunamis are described as a series of

waves, and solitary waves have long been employed as a

model for the leading wave of tsunamis. However, the effort

has been focused on the procurement of the best accessible

bathymetry/topography and preliminary situations in

constructing the most accurate model results. In a real world

situation, the authentication of a model is very imperative in

the process of ratifying an operational model. In this paper,

we acquired several numerical solutions to the KdV equation

and forced KdV equation by using LBM method for the

purpose of ratifying inundation models. The benefits of the

lattice Boltzmann method for forced KdV equation are of

greater flexibility in the choice of different forcing sources.

The proposed model’s effectiveness and precision is

authenticated using comprehensive numerical simulation,

which established that the LBGK results are in good

agreement with the analytical solution.

ACKNOWLEDGEMENT

This work was supported by a Special Education Program for

Offshore Plant by the Ministry of Trade, Industry and Energy

Affairs (MOTIE), South Korea.

REFERENCES

[1] Synolakis, C. E. 1986. The Runup of Long Waves. Ph.

D. Thesis, California Institute of Technology, Pasadena,

California, 91125, 288.

[2] Synolakis, C. E. 1987. The Runup of Solitary Waves. J.

Fluid Mech. 185: 523-545.

[3] Synolakis, C. E., Bernard, EN., Titov, VV., Kanoglu, U.,

and Gonzalez, FI. 2007. Standards Criteria and

Procedures for NOAA Evaluation of Tsunami Numerical

Models. NOAA Technical Memorandum OAR P,

Mel135:55.

[4] Hui Lin Lai, Chang Feng, M. A. 2009. A Higher Order

Lattice BGK Model for Simulating Some Nonlinear

Partial Differential Equations. Science in China Series

G; Physics, Mechanics & Astronomy. 52(7): 1053-1061.

[5] Chen, S., and Doolen, G. D. 1998. Lattice Boltzmann

Method for Fluid Flows. Annu. Rev. Fluid Mech. 30:

329-364.

[6] Guang, Y., and Zhang, J. 2009. A Higher Order Moment

Method of the Lattice Boltzmann Model for Korteweg-

de Vries Equation. Mathematics and Computers in

Simulation. 79: 1554-1565.

[7] Whitham, G. W., Chen, Y. S., and Hu, S. X. 1998. A

Lattice Boltzmann Method for KdV Equation. Acta

Mech. Sin. 14: 18-26.

[8] Holdych, D. J., Noble, DR., Georgiadis, J. G. et al. 2004.

Truncation Error Analysis of Lattice Boltzmann

Methods. J. Comput. Phys. 193: 595-619.

[9] Qiaojie, Li,, Zong, Ji, Zhoushun, Zheng and Hongjuan,

Liu. 2011. Numerical Solution of Nonlinear Klein-

Gordon Equation using lattice Boltzmann Method.

Applied Mathematics. 2: 1479-1485.

[10] Jun-Xiao, Zhao, and Bo-Ling, Guo. 2009. Analytic

Solutions to Forced KdV Equation. Journal of Chinese

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