Development of a Semi-Analytical Dynamic Force Model

MARIN AKTER1,2, MOHAMMAD ABDUL ALIM1, MD. MANJURUL

HUSSAIN3, KAZI SHAMSUNNAHAR MITA4, ANISUL HAQUE3, MD.

MUNSUR RAHMAN3, MD. RAYHANUR RAHMAN3

1Department of Mathematics, Bangladesh University of Engineering and Technology,

BANGLADESH

2School of Science, University of New South Wales, AUSTRALIA

3 Institute of Water and Flood Management, Bangladesh University of Engineering and

Technology, BANGLADESH

4Civil, Environmental and Ocean Engineering Department, Stevens Institute of Technology,

Hoboken, New Jersey 07030, USA

Abstract: - A moving water mass generates force which is exerted on its moving path. Cyclone generated storm

surge or earthquake generated tsunami are specific examples of moving water mass generated force along the

coasts. In addition to human lives, these moving water masses cause severe damages to the coastal infrastructure

due to tremendous force exerted on these structures. To assess the damage on these infrastructures, an essential

parameter is the resultant force exerted on these structures. To evaluate the damages, there is hardly any

quantitative method available to compute this force. In this paper we have developed a semi-analytical model,

named as Dynamic Force Model (DFM), by using Variational Iteration Method to compute this force. We have

derived the governing equation on the basis of Saint-Venant equations which are basically 1D shallow water

equations derived from the Navier-Stokes equations. DFM is verified, calibrated, validated, and applied in

Bangladesh coastal zone to compute dynamic thrust force due to tropical cyclone SIDR.

Key-Words: - Semi-analytical method, Variational Iteration Method, Moving water mass, Thrust force, Navier-

Stokes equations, Saint-Venant equations.

Received: June 19, 2023. Revised: February 23, 2024. Accepted: March 21, 2024. Published: June 3, 2024.

1. Introduction

A moving water mass generates force due to

momentum created by the water mass itself and the

acceleration due to its movement. Specific examples

of moving water mass are surge waves generated by

cyclones and tsunami waves generated by

earthquakes that strike the coast with considerable

magnitude of force. Though the sources of these

moving water masses (for example storm surge and

tsunami) are different, the physical characteristics of

their wave propagation (in deep water), nonlinear

transformation (in shallow water) and runup (land

inundation) are identical [1]. Most of the damages of

coastal infrastructures are due to this thrust force. To

assess stability of these structures and to ensure

safety of coastal populations, it is extremely

important to accurately compute this thrust force.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

The rapid changes in hydrodynamics near shore

substantially damage infrastructures of coastal

regions. So, from the major concepts of coastal

engineering, storm loading [2] (wind load, waves,

hydrostatic and hydrodynamic loads) and tsunami

induced forces are necessary in a coastal building

code. Water-related loads accompanying a coastal

storm are often not adequately addressed in coastal

building code, with wind generated/ induced waves

producing the most critical and complex forces to

which the coast and its structures are subjected [2].

Few current building codes of coastal structures such

as the city and county of Honolulu Building Code

(CCH); the 1997 Uniform Building Code (UBC 97);

the 2000 International Building Code (IBC 2000), the

SEI/ASCE 7-02 (ASCE 7), and the Federal

Emergency Management Agency Coastal

Construction Manual (FEMA CCM) incorporate the

effects of hydrodynamic loads (surge/tsunami

induced) [3]. The generalized equations of

contributing force components of these codes are

discussed in [1,4-5]. In these codes, computation of

hydrostatic force is the multiplicative combination of

gravitational acceleration and square of specific

energy. Hydrodynamic force is calculated as the drag

force acting on the direction of the uniform flow. But

these codes have over-simplified the process of

estimating the hydrodynamic loads [1]. The forces

are calculated for an instantaneous velocity of

moving water mass for design purposes of coastal

infrastructures. But these codes neglect the non-

linearity of the variation of the total force generated

during a cyclone or Tsunami.

A Tsunami induced force is done by Palermo et al,

2009 [5]. In their study, they considered three

parameters to define the magnitude and application

of tsunami-induced forces: (i) the inundation depth;

(ii) the flow velocity; and (iii) the flow direction [5].

But they did not consider the terms it may/must be

nonlinear of inundation depth, flow velocity and flow

direction as the parameters work together in the field.

Research Gap

From the synthesis of the past studies related to

computation of this thrust force, it is found that over

simplified empirical or semi-empirical approaches

are generally adopted mainly from engineering

perspectives by ignoring physics behind the

generation of this force. This force should actually be

calculated by solving the momentum equation by

considering acceleration of moving water mass and

wind stress due to cyclone effect.

A moving water mass generates force which is

exerted on its moving path that obeys always a

momentum equation. So, it is necessary to define the

thrust force that from the beginning of a cyclone, a

momentum equation should be considered. But no

such studies do this way. Rather they have calculated

the forces individually which are simpler. In reality,

the forces are highly nonlinear and they are

dependent on each other. During the derivation of

thrust forces from the momentum equation, few

nonlinear terms are generated which are not

considered in the previous studies.

Research Question

In this paper, it is discussed the following two

points: (a) How the thrust force generated by the

moving water mass due to cyclone or tsunami can be

computed by following a physics-based approach?

(b)What type of non-linear terms at the end can be

generated during the derivation of thrust force from

the momentum equation by considering moving water

mass and wind stress due to cyclone effect?

In this study, a physics-based approach is applied to

compute this force dynamically. Momentum

equation is used as the basic governing equation

which contains the force and acceleration

components of moving wind and water masses. The

momentum equation is based on the Saint-Venant

equation which is derived from the Navier-Stokes

equation. The equation is solved semi-analytically to

compute the force. As mentioned before, this force,

termed as dynamic thrust force, is responsible for the

damages of coastal infrastructures during

propagation of a cyclone and tsunami. The semi-

analytical model, named as DFM, developed in this

study can simulate this thrust force dynamically

during propagation of a cyclone or tsunami event.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

2. Materials and Method

2.1 Analytical Methods of Partial

Differential Equations

A wide variety of problems in fluid mechanics are

expressed by partial differential equations and they

are nonlinear. Nonlinearity exists everywhere, and

nature is nonlinear in general. There are many

methods, such as perturbation methods, to solve

nonlinear partial differential equations (PDEs). A

method known as the method of separation of

variables is perhaps one of the oldest systematic

methods for solving partial differential equations

including the wave equation. The wave equation and

its methods of solution attracted the attention of many

famous mathematicians including Leonhard Euler

(1707–1783), James Bernoulli (1667–1748), Daniel

Bernoulli (1700–1782), J.L. Lagrange (1736–1813),

and Jacques Hadamard (1865–1963) (Oliviera,

2021). They discovered solutions in several different

forms of partial differential equations. A well-known

analytical method is the Decomposition Method

which was established by Adomian. Special attention

should be paid to Adomian’s decomposition method

[6-7] and Liao’s homotopy analysis method [8]. With

these methods, most PDEs can be approximately

solved without linearization or weak linearization or

small perturbations. However, the approximation

obtained by Adomian’s method could not always

satisfy all its boundary conditions, leading to error

near boundaries. A successful approximation of

solution for partial differential equations is

established with no boundary problems by Ji-Huan

He [9-12] which is known as Variational Iteration

Method (VIM) [13]. This method is based on

analytically solving the equations that result from

discretizing the spatial coordinates of a partial

differential equation [14].

Governing equations formed based on Saint-Venant

equations are characteristically nonlinear partial

differential equations. Functional relations for some

of the variables in the governing equations are not

solvable analytically. Those relations are solved by

the discretization method. Here, Finite difference

method is used [15]. As the equations are solved by

applying coupled VIM that is analytical and

discretization methods which are known as

numerical. For this reason, in this study, this coupled

solution method is termed as ‘semi-analytical’.

2.2 Model Development

According to Newton's Second Law of Motion, the

change of momentum per unit of time is equal to the

result of all external forces applied to the moving

body. In this study, we have also considered wind

force exerted on the moving body. In this case, the

assumptions to the development of the governing

equations and its momentum equation is formed as

described below.

2.2.1 Assumptions to the development of Saint-

Venant Equations

In Fluid Dynamics, the Saint-Venant equations [16]

were formulated in the 19th century by two

mathematicians, Adhémar Jean Claude Barréde

Saint-Venant and Bousinnesque [16]. Saint-Venant

equations are derived from Navier-Stokes equations

[17] for shallow water conditions [18]. Navier-Stokes

equations represent a general model used to model

fluid flows [19-20]. On the other hand, a general

flood wave for 1-D situation [19-20] can be described

by the Saint-Venant equations.

During development of the Saint Venant equations,

following assumptions [17] were made:

 Flow is one-dimensional.

 Flow is unsteady and non-uniform.

 Hydrostatic pressure prevails and vertical

accelerations are negligible.

 Streamline curvature is small.

 Bottom slope of the channel is small.

 The fluid is incompressible.

 The gravity force is the only force which is

taken into account. So, the influence of the

Coriolis force is neglected.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

In addition to the above assumption, following

assumptions are made for this study:

 The steady uniform state of flow is expressed

by Manning’s equation which is used as the

initial condition for solution.

 Manning’s coefficient n is used to describe the

resistance effects.

 Wind is considered while forming the

momentum equation.

These three additional assumptions do not affect the

basic governing equation.

2.2.2. Conservation of momentum

Inside a control volume according to the conservation

of momentum, conservation of momentum can be

expressed (Vreugdenhil, 1994) as:



󰇟󰇠



󰇛󰇜











(1)

Here, the computation of Momentum flux, Pressure

force, Downslope gravitation force and Frictional

force in the equation (1) are taken from the studies

[16-17]. And the rest term, the frictional force on

water surface due to wind is calculated in this study

and it is described below.

If we consider  

 , the frictional force exerted on the

water surface due to cyclonic wind term is



 (2)

Where, the wind shear stress,  i.e.

wind force ; where uw is the wind

velocity, Cw is wind drag coefficient, B is width of

channel and  is the density of air.

Hence, in differential form, the momentum equation



󰇛󰇜

󰇛󰇜

 

 (3)

The gradient of the pressure force can be rewritten as



 





 (4)

Again, the ratio of the cross-sectional area A over the

wetted perimeter p, which has the dimension of a

length, called hydraulic radius and is defined as



(5)

Here, we consider wide water body, which is much

wider than the depth, the wetted perimeter is

generally not much more than the width 󰇛󰇜, and

so the hydraulic radius is approximately





(6)

Therefore, the equation (12) simplifies into













(7)

Equation (16) is called the Saint-Venant equation i.e.

1D shallow water equation.

If cyclonic wind is absent in the momentum equation,

the equation (7) becomes











(8)

It also implies a 1D shallow water equation for open

channel flow where cyclonic wind is absent.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

In present study, all these equations are solved semi-

analytically to compute spatial variation of forces

exerted by the wind and water mass.

2.2.3. Variational Iteration Method (VIM)

In 1978, Inokuti [21] proposed a general Lagrange

multiplier method to solve nonlinear problems. The

main feature of the method is as follows: the solution

of a mathematical problem with linearization

assumption is used as initial approximation or trial-

function, then a precise approximation at some

special point can be obtained. Considering the

following general nonlinear system,

󰇛󰇜󰇛󰇜󰇛󰇜

(9)

where L is a linear operator, N is a nonlinear operator

and g(t) is a known analytical function.

Above method was modified into an iteration method

[9-11, 22-23] in the following way:















nnnn dguNLuuu

1)()()(



(10)

where is a general Lagrange multiplier (Inokuti,

1978), which can be identified optimally via the

Variational theory [12,21, 23-24], the subscript n

denotes the nth approximation, and un is considered as

a restricted variation [23], i.e. 





The method is shown to solve effectively, easily, and

accurately a large class of non-linear problems with

approximations converging rapidly to accurate

solutions.

VIM is now widely used by many researchers to

solve linear and nonlinear partial differential

equations. The method introduces a reliable and

efficient process for a wide variety of scientific and

engineering applications, linear or nonlinear,

homogeneous or non-homogeneous, equations and

systems of equations as well. It was shown by many

authors [9-12; 25] that this method is more powerful

than existing techniques such as the Adomian method

[7, 26] or perturbation method [27]. The method

gives rapidly convergent successive approximations

of the exact solution if such a solution exists;

otherwise, a few approximations can be used for

numerical purposes. The existing numerical

techniques suffer from restrictive assumptions that

are used to handle nonlinear terms. The VIM has no

specific requirements, such as linearization, small

parameters, Adomian polynomials, etc. for nonlinear

operators. Another important advantage is that the

VIM method is capable of greatly reducing the size

of calculation while still maintaining high accuracy

of the solution. Moreover, the power of the method

gives it a wider applicability in handling large

numbers of analytical and numerical applications.

2.2.4. Solution of the Governing Equation

Equations (16) and (17) which are known as Saint

Venant equations can be re-written as













(11)

And 

 

 ℎ



ℎ

(12)

Where,  is air density, is water density,  is

cyclonic wind speed, u is water velocity, Cd is water

drag coefficient and Cw is wind drag coefficient.

We express the steady uniform state of flow by

Manning’s flow equation [28] as an initial condition

mentioned in the assumption for the Equations (11)

and (12), which means

󰇛󰇜





 (13)

where n is Manning’s roughness coefficient and Rh is

hydraulic radius of the channel.

The correction functional for Equation (11) is

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

󰇛󰇜󰇣

 

 











󰇤 (14)

The stationary condition is given by

 

0





 

01 



t





(15)

Equation (15) gives  and substituting the

value of the Lagrange multiplier into the correction

functional, Equation (14) gives the iteration formula.

󰇣

 

 











󰇤

(16)

The zeroth approximations

 

0, SR

xu h



are selected by using the given initial conditions.

Following are the successive approximations,

󰇛󰇜





 (17)

󰇣

 

 











󰇤

(18)









󰇩













































󰇪 (19)



ℎ



ℎ





ℎ󰇡

ℎ



󰇢

ℎ

(20)

as 

󰇡





󰇢

󰇡





󰇢

is the steady and uniform condition.

󰇣

 

 ℎ









ℎ

ℎ󰇤

(21)

Using Equation (20), we get the 2nd iterated formula



ℎ







ℎ



󰇭ℎ









ℎ



 

 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮

󰇭ℎ



ℎ



 

 



 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024



ℎ



󰇭ℎ









ℎ





ℎ



 󰇮



ℎ󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ





(22)

Definite integral

If a function f(x) is continuous on the interval [a, b]

which is divided into n subintervals of equal width 

, the definite integral of f(x) from a to b is

󰇛󰇜

∞󰇛󰇜









(23)

Using definite integral, we have

















and







 (24)

Equation (22) is simplified into



ℎ







ℎ



󰇭ℎ









ℎ



 

 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮

󰇭ℎ



ℎ



 

 



 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ



󰇭ℎ









ℎ





ℎ



 󰇮



ℎ󰇡ℎ



ℎ







ℎ

 󰇢

ℎ



(25)

Equation (25) is the approximate solution of Saint-

Venant equation (Equation (11)) to compute velocity

u (including water depth and wind speed), i.e. 

Therefore,



ℎ





ℎ



󰇭ℎ









ℎ



 

 󰇮

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

󰇭ℎ



ℎ





 

ℎ



 󰇮

󰇭ℎ



ℎ



 

 



 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ



󰇭ℎ









ℎ





ℎ



 󰇮



ℎ󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ





(26)

If we do not consider any wind term in the

momentum equation, the solution becomes



ℎ





ℎ



󰇭ℎ





 󰇮



󰇭ℎ





 

ℎ

󰇮󰇭ℎ





 󰇮



󰇭ℎ



ℎ





 󰇮



ℎ



󰇭ℎ









ℎ

󰇮



ℎ󰇡ℎ



ℎ



 󰇢

(27)

Equation (27) is the approximate solution of Saint-

Venant equation (when wind speed is absent) given

by Equation (12).

2.3 Computation of Dynamic Thrust Force

Let u(x, t) denotes the water velocity (computed by

using water depth and wind speed as input) at spatial

location x and temporal state t.

If a water particle travels along the curve through the

water body, the rate of change of velocity creates two

kinds of accelerations: (a) Local acceleration and (b)

Convective acceleration. It is worth mentioning that

a force produced from the moving body affected by

both wind and water mass is called thrust force in this

study.

The acceleration for water particle can be calculated

by using







(28)

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

where, 

 is called local acceleration i.e temporal

variation and 

 is called convective acceleration

when the particles move through regions with

spatially varying velocity.

Since the water under consideration is moving, it is

acted upon by external forces which will follow

Newton’s second law. i.e.

Therefore, the force for moving fluid (Yazar, et.al.,

2024) can be calculated as



















x



(29)

which is considered as the dynamic thrust force due

to cyclone generated surge wave.

Using Equation (26), we can compute the local

acceleration as





ℎ



󰇭ℎ









ℎ

 



 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮

󰇭ℎ



ℎ

 







 󰇮

ℎ



ℎ





ℎ



ℎ



󰇭ℎ









ℎ





ℎ



 󰇮



ℎ󰇡ℎ



ℎ







ℎ

 󰇢

(30)

Again, using Equation (26), we can compute the

convective acceleration as





ℎ



󰇭ℎ









ℎ



 





ℎ





 󰇮

󰇭ℎ









ℎ



 

 󰇮󰇭ℎ









ℎ



 

 󰇮

󰇭ℎ



ℎ



 





 

ℎ





 󰇮

󰇭ℎ





ℎ









 

ℎ



 󰇮

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

󰇭ℎ



ℎ



 

 



 󰇮



ℎ





󰇭ℎ



ℎ



 

 



 󰇮



ℎ󰇭ℎ





 



ℎ









ℎ



 󰇮

󰇭ℎ



ℎ



 

 



 󰇮

(31)

Now we can write the Equation (29) as

󰇯󰇱

ℎ



󰇭ℎ









ℎ

 



 󰇮

󰇭ℎ



ℎ









ℎ



 󰇮󰇭ℎ









ℎ

 



 󰇮

ℎ



ℎ



ℎ



ℎ





󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ





󰇭ℎ



ℎ









 

ℎ





 󰇮

󰇭ℎ









ℎ

 



 󰇮󰇭ℎ









ℎ

 



 󰇮

󰇭ℎ



ℎ









 

ℎ





 󰇮

󰇭ℎ



ℎ



ℎ



 󰇮

󰇭ℎ









ℎ

 



 󰇮

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024



ℎ



󰇭ℎ









ℎ

 



 󰇮

ℎ

󰇭ℎ



ℎ









ℎ



 󰇮󰇭ℎ









ℎ

 



 󰇮󰇲󰇰

(32)

Where,



ℎ





ℎ



󰇭ℎ









ℎ



 

 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮

󰇭ℎ



ℎ



 

 



 󰇮

󰇭ℎ



ℎ





 

ℎ



 󰇮



ℎ



󰇭ℎ









ℎ





ℎ



 󰇮



ℎ󰇡ℎ



ℎ







ℎ

 󰇢

ℎ



(33)

Equation (32) is the required equation to calculate

the dynamic thrust force due to momentum created

by the wind and water mass. In present study, these

are used to compute the dynamic thrust force exerted

by the cyclone generated storm surge. The model

represented by Equations (32) and (33) is named as

Dynamic Force Model (DFM).

Equations (32) and (33) contain the derivatives 

,



 ,

 and 

 that are solved numerically using

finite difference method [15] and their discretized

forms are written as:

ℎ

ℎℎ

 (34)



 

 (35)

ℎ

ℎℎℎ

 (36)





 (37)

Here

,.....3,2,1i

. Though the governing equation

of the DFM model is solved analytically

(Variational Iteration Method), some forms are

computed numerically (Finite Difference Method).

For this reason, it is termed as semi-analytical

model.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

To make the DFM directly coupled with Delft 3D

[29], water depth h, wind velocity uw and

discretization interval ∆x computed from Delft 3D

are directly inserted in Equations (32) and (33) t is

to be noted here that any numerical model in lieu of

Delft 3D can be coupled with DFM.

3. Model Verification

Model verification essentially means “model

algorithm and code are correctly implemented” [30].

In this study, the solution algorithm and model code

are verified by comparing the semi-analytical model

results with the numerical solution. We use the

Variational Iteration Method as an analytical

method. To verify this semi-analytical solution, the

numerical method selected is the Finite Difference

Method [15]. The reflexive condition is used as

boundary conditions during the numerical solution.

Using MATLAB, we have prepared two scripts for

the semi-analytical solution and the numerical

solution separately. During comparison, only the

velocity computed from the DFM is compared with

the numerical solution of the governing equation

from which the velocity field is also calculated. This

essentially serves the purpose of model verification.

3.1 Numerical solution using finite difference

method

Finite difference method [31] is considered as a

numerical method for the solutions of the governing

equation (11).

To verify the DFM algorithm and code, the velocity

field computed by the Equation (26) is compared

with the velocity field computed from numerical

solution of the governing equation (11) in a

hypothetical channel where water is flowing from

upstream to downstream.

3.1.1 Hypothetical channel description

A hypothetical channel of 5.1km long, 100m width

and 5m deep is considered for verification of the

model (Fig. 1). Winds with variable speeds are

assumed to blow over the channel. Steady and

uniform state of flow represented by Manning’s

equation i.e. 

 is used as initial

condition, where  is the hydraulic radius and  is

the bed slope of the channel.

Fig. 1: Hypothetical channel

3 different cases (Case-1, Case-2 and Case-3) are

constructed to compare the velocities computed by

DFM with the numerical solution. In Case-1, water

depth and water surface slope are kept constant (Fig.

2) with no wind blowing over the channel. This

results in almost constant water velocity along the

channel. Computed water velocities from the DFM

and the numerical solution are shown in Fig. 3. The

result shows a near perfect agreement between the

DFM and the numerical solution with R2=0.96.

Fig. 2: Water depths h(m) in different positions

x(km) along the horizontal for Case-1.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Fig. 3: Comparison of longitudinal velocity profile

u(m/s) between the DFM and the numerical solution

for Case-1.

In Case-2, water depth is varying along the channel,

but no wind is blowing (Fig. 4). This results in

variable water surface gradients along the channel.

Comparison of computed water velocity between the

DFM and the numerical solution is shown in Fig. 5.

The result shows a slight decrease of agreement

between the DFM and the numerical solution

compared to Case-1 (R2 decreases from 0.96 to 0.91).

Fig. 4: Water depths h(m) in different positions

x(km) along the horizontal for Case-2.

Fig. 5: Comparison of longitudinal velocity profile

u(m/s) between the DFM and the numerical solution

for Case-2.

In Case-3, wind with variable speed is assumed to

blow over the channel where water depth was

initially constant (Fig. 6). Variable wind speed results

change in water surface gradient along the channel.

Comparison of water velocity between the DFM and

the numerical solution for Case-3 is shown in Fig. 7.

The result shows further decrease of agreement

between the DFM and the numerical solution

compared to Case-1 and Case-2 (R2=0.82 in Case-3

compared to 0.96 in Case-1 and 0.91 in Case-2).

The verification results show systematic deviation of

numerical solution from the analytical solution with

the increase of non-linearity in the system, which is

an inherent drawback of numerical solutions. This

shows that both the solution algorithm and code are

appropriately implemented in DFM.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Fig. 6: Variable wind Speed uw(m/s) in different

positions x(m) along the channel for Case-3.

Fig. 7: Comparison of longitudinal velocity profile

u(m/s) between the DFM and the numerical solution

for Case-3.

4. Model Calibration

By model calibration, we generally mean selecting

the appropriate model parameters that represent

experimental or field conditions [30]. DFM has two

parameters that need to be calibrated. These are: wind

drag coefficient Cw (see Equation 11) and

Manning’s n (see Equation 12). As mentioned

before, the discretized Equations (34)-(37) are solved

by coupling the DFM with Delft3D. The field

application of DFM is made by coupling the field

simulation result of Delft3D. To make DFM

compatible with Delft3D, the same Manning’s n is

used both for DFM and Delft3D. As the solution

algorithm between DFM and Delft3D is different, for

wind drag coefficient Cw, an empirical equation is

derived between Cw and wind velocity uw which is

used for different field applications of DFM. To

derive this empirical equation, values of the wind

drag coefficient used in a particular application of

Delft3D (which is 0.002) and simulated water

velocity from the same application of Delft3D is

used. To make values of Cw used in Delft3D

compatible with the DFM, following empirical

equation is used:















C002.0

(38)

Where, Cw is a wind drag coefficient used as the

calibration parameter of DFM, u is the simulated

water velocity for a particular application of Delft3D

and ud is the computed velocity for the same

application of DFM. After a series of model runs with

DFM, relation between wind drag coefficient Cw and

wind velocity uw is established and is shown in Fig.

Fig. 8: Relation between wind drag coefficient Cw

and wind velocity uw computed by applying DFM

Using the relation in Fig. 8, following empirical

equation between Cw and uw is developed:

Cw = 0.000004uw2 + 0.002

(39)

During subsequent model application by DFM,

Equation (39) is used to compute the wind drag

coefficient for specific cyclone wind speed

represented by uw.

The transfer of momentum from air to the ocean

surface produces a wave field that imposes shear

stress which ultimately causes the rise of water level

[32]. The drag coefficient Cw controls the transfer of

momentum from air to water surface in the governing

momentum equation [33]. It mainly depends on wind

speed. In modeling storm surge, this coefficient is a

crucial parameter [34]. Past studies show that this co-

efficient either remains constant or increases with the

increase of wind speed [35-40]. In the present study,

it is found that the wind drag coefficient has a

nonlinear relation with the wind speed (see Fig. 8 and

Equation (50)).

5. Model Validation

Model validation means the ability of the model to

represent the real world for which the model is

developed [30]. As real-world data of thrust force for

any moving water mass is not available, validation of

DFM is performed from two different perspectives.

These are:

R² = 0.727

0,005

0,01

0,015

0,02

020 40 60 80

Drag Coefficient (Cw)

Wind Speed (uw)

Drag Coefficient (Cw)

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

1. Validation of thrust force of DFM by

comparing velocity-force curve generated

for tsunami and cyclonic storm surge.

2. Validation of flow field of DFM with flow

field simulation from calibrated and

validated numerical model Delft 3D.

5.5 Validation with the tsunami force

Tsunami is a small amplitude wave created by lifting

of water mass above the sea bottom due to rapture of

the sea floor created mainly by earthquakes [41]. The

potential energy of this displaced water mass

converts into kinetic energy [41] and propagates with

tremendous velocity (as high as 800 km/hr) in deep

sea. This wave, when it reaches the coast, is slowed

down but amplified due to shallow water effect and

strikes the coastal structures with the same energy it

had when it was created in the deep sea. On the other

hand, cyclone generated storm surge is an elevated

sea surface above the tide level due to lower

atmospheric pressure and cyclonic wind. The storm

surge acts simultaneously with tide and wind waves.

The energy created due to storm surge is proportional

to the tidal range of storm tide [42]. Similar to

tsunamis, the storm surge is amplified near the coast

due to the shallow water effect. The tsunami wave

contains energy due to displacement of sea water

mass above the sea-bed, whereas the storm surge

contains energy created due to difference of sea

surface elevation measured by tidal range of storm

tide [41-42]. Energy of a tsunami wave is several

times higher than a storm surge (water depth above

sea-bed can be as high as 4000m, whereas, tidal range

of a storm tide can vary between 6m to 7m depending

on the tidal amplitude and strength of the cyclone).

After reaching the coast, both the tsunami wave and

storm surge strikes the coastal structure with a force

which is directly proportional to the energy [43]. This

means that after reaching the coast, the water mass

created due to tsunami and moving with a certain

velocity will exert more force on a coastal structure

compared to a moving water mass moving with the

same velocity but created due to cyclonic storm

surge. The reason is – the moving water mass created

by tsunami contains more energy compared to the

moving water mass created by storm surge [41-42]

and force is directly proportional to energy [43].

In the validation exercise of DFM force, we generate

velocity-force curves for both tsunami and storm

surge. For tsunamis, the velocity-force curve is

generated by using the relation developed by Murata,

et al. (2011) [44] based on Japan tsunami study. They

expressed the water velocity during tsunami wave

propagation as u and the drag force of tsunami as Fd

acting on coastal structures as:

 (40)

 (41)

Where,  is the specific weight of seawater,  is a

drag coefficient, h is water depth and B is the breadth

of a building in the flow direction. Velocity-force

curve for tsunami wave is generated by calculating

force with the following parameter values suggested

by Murata, et al. (2011) [44]:

 = 10.03 kN/m3

 = 2.2

 = 1.0m to compute the force per unit length of the

structure.

Values of  in Equation (41) is calculated from

Equation (40) by assuming different ranges of

velocity 

For storm surge, the velocity-force curve is generated

by applying DFM in the hypothetical channel which

was used for model verification (Fig. 1). In this case,

we used case-3 model setup (Fig. 6), but to get a

reasonable range of force to generate velocity-force

curve, we have used a higher range of wind speed

compared to the wind speed range shown in Fig. 6.

To generate velocity-force curves, different ranges of

velocity are computed in different sections of the

hypothetical channel by applying DFM. These

velocities are used in Equation (29) to compute the

force numerically. The numerical solution of

Equation (29) is equivalent to the semi-analytical

solution of Equation (32).

Comparison of velocity-force curve between tsunami

and storm surge is shown in Fig. 9. The results show

that for the same velocity, tsunami force is several

times higher than storm surge force. With increasing

velocity, this difference also increases. We have

already discussed why tsunami force should be

higher than storm surge force. The velocity-force

curve in Fig. 9 shows that DFM realistically

computes force due to cyclonic storm surge.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Fig. 9: Comparison of velocity-force curve between

tsunami and storm surge.

5.6 Validation with Delft3D simulation

In this validation approach of DFM, the velocity field

simulated by DFM is compared with the velocity

field simulated by Delft 3D. The study area for this

comparison is the coastal zone of Bangladesh (Fig.

10) for the tropical cyclone event SIDR that made

landfall on the Bangladesh coast on November 15,

2007. Comparison of surge velocities between Delft

3D and DFM at the time of landfall is shown in Fig.

11.

Fig. 10: Coastal zone of Bangladesh.

Fig. 11: Comparison of surge velocities between

DFM and Delft 3D.

To quantify the visual qualitative color comparison,

a one-to-one function is defined among the colors

between the two maps. In this way, it is possible to

quantify the overlapped color and deviated color

between the two maps. The overlapped color shows

the similarity, and the deviated color shows the

dissimilarity between the maps. The comparison is

shown in Table 1. The comparison shows that surge

velocities computed by DFM is 97% similar to that

computed by Delft3D. It is noted here that Delft3D

applied in this study is a validated storm surge model

in the coastal zone of Bangladesh [45]. From this

perspective, this result shows the field validation of

one important parameter of DFM.

Table 1: Map Comparison

Models

In Percentage (%)

Overlapped

Colors

Deviated

Colors

Delft3D – DFM

97%

6. Model Application

The DFM model has numerous prospects in

improving coastal resilience in particular and

structural resilience in general. The model is not only

able to compute dynamic thrust force due to cyclone

generated storm surge, but also can compute the

dynamic thrust force exerted by any moving water

150

300

450

600

750

900

100

200

300

400

500

600

2 7 12

Tsunami Force(kN/m)

Cycloninc Force(kN/m)

Velocity(m/s)

Cyclonic Force

Tsunami Force

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

mass on a structure in its flow path (for example due

to tidal wave, wind wave, flows in ocean, estuaries

and rivers, tsunami etc.). The output from the model

can be used as one of the design criteria of

infrastructures on which force is exerted by the

moving water mass. As the DFM is capable of

calculating spatial and temporal changes of thrust

force, it can also be considered as one of the

monitoring parameters of extreme climatic

conditions related to moving water. As an

illustration, the model is applied to compute dynamic

thrust force due to tropical cyclone SIDR.

Tropical cyclone SIDR is considered as one of the

devastating cyclones that made landfall on November

15, 2007 at the east of Sundarban in Bangladesh

coast. During the time of landfall, the maximum wind

speed of this cyclone was 215 km/hr [46]. This

cyclone is believed to generate a maximum surge

depth of 6m [47-48]. Simulated thrust force due to

cyclone SIDR at the time of landfall is shown in Fig.

12. Table 2 shows the maximum thrust force at

different coastal districts in Bangladesh (the district

names are shown in Fig. 12) due to cyclone SIDR.

Table 2: Maximum thrust force at different coastal

districts of Bangladesh due to cyclone SIDR.

District

Maximum thrust

force (F) in kN/m

Patuakhali

110.5

Barguna

86.6

Bhola

54.1

Pirojpur

49.6

Jhalokati

44.5

Barisal

40.2

Bagerhat

33.2

Noakhali

17.9

Khulna

8.4

Chittagong

6.9

Lakshmipur

6.3

Cox's Bazar

4.8

Satkhira

4.3

Shariatpur

3.1

Narail

2.4

Gopalganj

2.2

Chandpur

2.1

Feni

1.9

Jessore

0.3

According to simulation results of DFM, the coastal

districts of Patuakhali, Barguna, and Bhola (see Fig.

12) are impacted with a very high magnitude of thrust

force (magnitude greater than 50 kN/m, see Table 2).

In the post-disaster damage and loss assessment

report, these districts are also identified as the worst

affected coastal districts in terms of economic and

infrastructure loss [47]. Due to anti-clockwise

rotational impacts of cyclones in the northern

hemisphere, locations on the right side of cyclone

tracks are the most affected. According to DFM

simulation, coastal districts of Patuakhali, Barguna

and Bhola are the worst affected regions which are

situated at the right side of the cyclone track and at

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

the same time close to the path of the cyclone track.

On the other hand, although Pirojpur, Jhalokati and

Bagerhat are situated close to the cyclone track (see

Fig. 12), magnitude of thrust forces are relatively less

in these districts (less than 50 kN/m) because these

districts are located at the left side of the cyclone

track. The red lines in Fig. 12 show the coastal

embankments (locally known as polders) to protect

the lands from surge waves. DFM simulated thrust

force inside the protected land is only due to cyclone

wind, not due to combined impact of cyclone wind

and moving surge. But the thrust force in the

unprotected land is due to combined impacts of

moving surge and cyclone wind. Thrust force

simulated by DFM can be applied to assess

probability of failure of embankments during

propagation of surge waves and possible damage of

households both inside and outside the protected

land. These data can also be used to further

strengthen these structures or during construction of

new structures in the coastal

region.

Fig. 12. Computed dynamic thrust force due to

cyclone SIDR at the time of landfall in the coastal

zone of Bangladesh.

7. Conclusion

A Dynamic Force Model (DFM) is developed by

semi-analytically solving the Saint-Venant

equations. The model can dynamically compute the

thrust force due to any moving water mass including

cyclone generated storm surge. As an analytical

method, Variational Iteration Method (VIM) is used

to solve the equations. During solution, steady and

uniform state of flow computed by Manning’s

equation is considered as an initial condition. Thrust

force is computed from local and convective

accelerations of moving water.

The model is verified (verification of solution

algorithm and model code) by applying the model in

a hypothetical channel on which wind is blowing.

The computed water velocity by semi-analytical

solution is compared with a finite difference solution.

The verification results show systematic deviation of

numerical solution from the analytical solution with

increasing non-linearity in the system, which is an

inherent drawback of numerical solutions. This

shows the model algorithm and code are

appropriately implemented. To apply the model in

simulating dynamic thrust force due to cyclone

generated storm surge, Manning’s n and wind drag

coefficients are used as the calibration parameters.

For the DFM, an empirical equation is developed by

relating wind drag coefficient with wind speed which

can be used for any specific application (related to

wind driven water flow) of the model. It is found that

the wind drag coefficient increases non-linearly with

the increase of wind speed. Thrust force computed by

DFM is validated by comparing the model result with

the tsunami data studied in Japan. On the other hand,

surge velocity computed by DFM is validated by

comparing the model result with the surge velocity

computed by field validated numerical model Delft

3D. The verified, calibrated and validated DFM is

applied to simulate dynamic thrust force in the

Bangladesh zone due to tropical cyclone SIDR. This

specific application shows the importance of thrust

force as a parameter to assess performance of coastal

infrastructure. DFM has the potential to be applied in

engineering stability analysis of coastal

infrastructures (for example embankments and

building structures) to make these structures resilient

against cyclone generated storm surge and tsunami.

Next version of DFM will include the following:

1. DFM can be made independent by

analytically solving the mass and momentum

equations to compute the surge depth.

2. The zero-surge-depth problem needs to be

solved mathematically.

3. DFM will be made a coupled module to the

present version of Delft 3D. As Delft3D is

open source, this coupling will make DFM to

be used globally as part of Delft3D.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Acknowledgement:

Collecting the bathymetry data for Delft3D model

was supported by the ‘ESPA Deltas (NE/J002755/1)’

project funded by the ESPA programme, through

Department for International Development (DFID),

the Economic and Social Research Council (ESRC)

and the Natural Environment Research Council

(NERC), UK, and the National Water Resources

Database hosted at Water Resources Planning

Organization (WARPO), Ministry of Water

Resources (MoWR), Government of the People's

Republic of Bangladesh (GoB). The model

development was supported by the DECCMA Project

(Grant No. IDRC 107642), part of the Collaborative

Adaptation Research Initiative in Africa and Asia

(CARIAA), with financial support from the UK

Government's DfID and the International

Development Research Centre (IDRC), Canada, the

WARPO Projects, ‘Research on the Morphological

processes under Climate Changes, Sea Level Rise

and Anthropogenic Intervention in the coastal zone’

and ‘Research on Sediment Distribution and

Management in South-West Region of Bangladesh’

funded from MoWR, GoB and SATREPS

(0510000000023) (funded by JST-JICA) through

‘Research on Disaster Mitigation against Strom

surges and floods in Bangladesh’ are gratefully

acknowledged.

This work is a part of author's Master's thesis, which

is supported by DECCMA project which is further

fine-tuned through different projects including the

recent fellowship project in 2023 by Coalition for

Disaster Resilient Infrastructures (CDRI). The views

expressed in this work are those of the creators and

do not necessarily represent those of DFID and

IDRC, JICA, WARPO, CDRI or its Board of

Governors

References:

[1] Solomon, Y. Modeling and simulation of tsunami

and storm surge hydrodynamic loads on coastal

bridge structures, 21st US-Japan Bridge Engineering

Workshop, 2006.

[2] Clark, R. R. Structural design aspects of a coastal

building code. Coastal Engineering Proceedings,

1(18),1982.

[3] Pacheco, K.H. and Robertson, I.N., Evaluation of

Tsunami Loads and other effect on Reinforced

Concrete Buildings, http://www.cee.hawaii.edu/wp-

content/uploads/UHM-CEE-05-06.pdf, 2005.

[4] Yeh, H., & Robertson, I. N. Development of

design guideline for tsunami shelters. In Proc., First

International Conference on Urban Disaster

Reduction (1ICDR) (p. 2005)

[5] Palermo, D., Nistor, I., Nouri, Y., & Cornett, A.

Tsunami loading of near-shoreline structures: a

primer. Canadian Journal of Civil Engineering,

36(11), 1804-1815, 2009.

[6] Adomian, G., Stochastic System, New York :

[7] Adomian, G., A Review of the Decomposition

Method in Applied Mathematics. J. Math. Anal.

Applic, 135:501-544, 1988.

[8] Liao, S., Homotopy analysis method: A new

analytical technique for nonlinear problems” J. of

Communications in Nonlinear Science and

Numerical Simulation,

https://doi.org/10.1016/S1007-5704(97)90047-2,

1997.

[9] He, J.H. Variational iteration method for delay

differential equations, Commun. Non-linear Sci.

Numer. Simulation 2 (4) (1997) 235—236.

[10] He, J.H. Variational iteration approach to 2-

spring system, Mech. Sci. Technol. 17(2) (1998)

221—223.

[11] He, J.H. Non-linear Oscillation with Fractional

Derivative and its Approximation, Int. Conf. on

Vibration Engineering 98, Dalian, China, 1988.J.-H.

He / International Journal of Non-Linear Mechanics

34 (1999a) 699–708 707.

[12] He, J.H. Variational Iteration Method- a kind of

non-linear analytical technique: some examples,

International journal of non-linear Mechanics. Vol.

34, 1999b, pp.699-708

[13] Sinha, V. K., and Maroju, P.. New Development

of Variational Iteration Method Using

Quasilinearization Method for Solving Nonlinear

Problems, Mathematics 11, no. 4: 935 (2023)

https://doi.org/10.3390/math11040935

[14] Vidts, P.D. and White R.E. A Semi-Analytical

Solution Method for Linear Partial Differential

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Equations, Computer and Chemical Engineering,

Vol. 16, No. IO/II, 1992, pp. 1007-1009.

[15] Mustapha, K.A., Furati, K.M., Knio, O.M. A

Finite Difference Method for Space Fractional

Differential Equations with Variable Diffusivity

Coefficient. Commun. Appl. Math. Comput. 2, 671–

688 (2020). https://doi.org/10.1007/s42967-020-

00066-6

[16] Raisinghania, M.D., Fluid Dynamics, chapter-

12, p-928-930. India. 2003.

[17] Vreugdenhil, C.D. Numerical Method for

Shallow water equations. Boston: kluwer Academic

publishers, 1994.

[18] Kubatko, E., Development, Implementation and

Verification of hp- Discontinuous Galerkin Models

for shallow Water Hydrodynamics and Transport,

PhD. Dissertation, 2005.

[19] Bessona, O., Kaneb, S. and Syc, M., On a 1D-

Shallow Water Model: Existence of solution and

numerical simulations, International Conference in

Honor of Claude Lobry, 2007.

[20] Bulatov, O. V. Analytical and Numerical

Riemann Solutions of the Saint Venant Equations for

Forward and Backward Facing Step Flows,

Computational Mathematics and Mathematical

Physics, Faculty of Physics, Moscow State

University, Moscow, 10.1134/S0965542514010047,

2013.

[21] Inokuti, M. General use of the Lagrange

multiplier in non-linear mathematical physics, in: S.

Nemat-Nasser (Ed.), Variational Method in the

Mechanics of Solids, Pergamon Press, Oxford, 1978,

pp. 156—162.

[22] Mungkasi, S., & Hari Wiryanto, L. Variational

Iteration Solution to the Gravity Wave-Model

Equations. Journal of Physics: Conference Series,

1007, 2018.

[23] Finlayson, B.A., The Method of Weighted

Residuals and Variational Principles, Academic

Press, New York, 1972.

[24] Nayfeh, A.H., Problems in Perturbation, Wiley,

New York, 1985.

[25] Abdou, M.A. and Soliman, A.A. New

Application of Variational Iteration Method,

Nonlinear Phenomena, 211,

10.1016/j.physd.2005.08.002, 2005, pp.1-8.

[26] Hagedorn, P. Non-linear Oscillations (translated

by Wolfram Stadler), Clarendon Press, Oxford, 1981.

[27] Holmes M.H. Perturbation Methods. In:

Introduction to the Foundations of Applied

Mathematics. Applied Mathematics, Springer, New

York, NY, 2009.

[28] Chow, V. T. Open-channel hydraulics.

McGraw-Hill Book Co., New York, N.Y., 1959.

[29] Deltares, Delft3D-Flow, Simulation of multi-

dimensional hydrodynamic flows and transport

phenomena, including sediments, User Manual,

Version 3.15.34158, May 2014.

[30] Trucano,_T.G., Swiler,L.P., Igusa,T.,

Oberkampf, W.L. and Pilch, M., Calibration,

validation, and sensitivity analysis: What’s what,

Reliability Engineering and System Safety, 91 (2006)

1331–1357.

[31] Lundgren, L. and Mattsson, K. An efficient

finite difference method for the shallow water

equations. Journal of Computational Physics, 422(),

109784, doi:10.1016/j.jcp.2020.109784

[32] Powell, M.D., Vickery, P.J. and Reinhold, T.A.,

Reduced drag coefficient for high wind speeds in

tropical cyclones, Nature, vol. 422, March 20. 2003,

pp.279-283.

[33] Donelan, M.A., Drennan, W.M. and Katsaros,

K.B. The Air–Sea Momentum Flux in Conditions of

Wind Sea and Swell, Journal of Physical

Oceanography, Vol(27), 1996, 10.1175/1520-

0485(1997)027.

[34] Drews, C.W., Using wind set down and storm

surge on Lake Fire to calibrate the air sea drag

coefficient, PLoS One, 8,

101371/Journal.pone.0072510, 2013.

[35] Charnock, H., Wind stress on a Water Surface,

Quart. J. Roy. Meteorol. Soc., 81,1955, 639-640.

[36] Large, W.G. and Pond, S., Open Ocean

Momentum Flux Measurements in Moderate to

Strong Winds, J. Phys. Ocean ogr., 11, 1981, pp.324-

336.

[37] Smith, S.D. and Banke, E.G., Variation of the

sea surface drag coefficient with wind speed. Quart,

J. Roy.Meteorol.Soc., 101, 1975, pp.665-673.

[38] Bender, M.A., Ginis, I., Tuleya, G.R., Thomas,

B. and Marchok, T., The operational GFDL Coupled

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Huricane-Ocean Prediction System and a Summery

of its performance, 3966, Mon. Wea. Rev., 135,

2007, 3965-3989.

[39] Kohno, M. and Higaki, M., The Development of

a Storm Surge Model including the Effect of Wave

set up for Operational Forecasting Meteorology and

Geophysics. Papers in Meteorology and Geophysics,

57, 2006, 11-19, doi:10.2467/mripapers.57.11.

[40] Zhizhua, Z., Wang, Y., Yihong, D., Lianshou,

C., Zhiqiu, G., On Sea Surface Roughness

Parameterization and Its effect on tropical Cyclone

Structure and Intensity. Advances in Atmospheric

Sciences, vol. 27, no.1-4, 2010. pp.98-117.

[41] Dutykh Denys and Frederic Dias, Energy of

tsunami waves generated by bottom motion, Proc. R.

Soc. A (2009) 465, 725–744,

doi:10.1098/rspa.2008.0332.

[42] Lewis, M.J. , Angeloudis, A., Robins, P.E. ,

Evans, P.S. and Neill, S.P. Influence of storm surge

on tidal range energy, Energy, Volume 122, 2017,

Pages 25-36, 2017, 0360-5442.

[43] Oetjen, J., Sundar, V., Venkatachalam, S. A

comprehensive review on structural tsunami

countermeasures. Nat Hazards 113, 2022, 1419–

1449, https://doi.org/10.1007/s11069-022-05367-y

[44] Murata, S., Imamura, F., Katoh, K., Kawata, Y.,

Takahashi, S. and Takayama, T. TSUNAMI to

survive from Tsnuami, Advanced Series on Ocean

Engineering, Vol (32), 2011.

[45] Haque, A., Kay, S. and Nichols, R.J. Present and

future fluvial, tidal and storm surge flooding in

coastal Bangladesh. Ecosystem Services for Well-

Being in Deltas, Palgrave,2018.

[46] Islam, A.K., Haque, M.S. and Bala, S.K.

Hydrologic Characteristics of Floods in the Ganges-

Brahmaputra-Meghna (GBM) delta, Natural

Hazards, 54(3): 2010, 797-811. DOI:

10.1007/s11069-010-9504.

[47] GoB, Cyclone Sidr in Bangladesh: Damage,

Loss and Needs Assessment for Disaster Recovery

and Reconstruction. MoFDM, Dhaka, Bangladesh,

2008.

[48] Sakib, M., Nihal, F., Akter, R., Maruf, M.,

Akter, M., Haque, A., Rahman, M. ―Afforestation

as a Buffer against Storm Surge Flooding along the

Bangladesh Coastal. 12thInternational Conference

on Hydroscience & Engineering, 2016.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Marin Akter (Corresponding Author); She had

major role to complete the task. Especially in

Model development, validation, calibration and

verification.

Mohammad Abdul Alim; He contributed on

supervising.

Md. Manjurul Hossain; He contributed on

programming on numerical code.

Kazi Samsunnahar Mita; she contribute on

introduction writing.

Anisul Haque; he contributed on mathematical

equations.

Munsur Rahman; He contributed on supervising.

Md. Rayhanur Rahman; He contributed on

Introduction writing.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Funding is not applicable

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2024.6.6

Marin Akter, Mohammad Abdul Alim,

Md. Manjurul Hussain, Kazi Shamsunnahar Mita,

Anisul Haque, Md. Munsur Rahman, Md. Rayhanur Rahman

E-ISSN: 2766-9823

Volume 6, 2024

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US