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Storm surge simulation along the Meghna estuarine area: an alternative approach
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Gour Chandra PAUL1, *, Sukumar SENTHILKUMAR2, Rana PRIA3
Acta Oceanologica Sinica | 2018, 37(1) : 40 - 49
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Acta Oceanologica Sinica | 2018, 37(1): 40-49
• Physical Oceanography, Marine Meteorology and Marine Physics •
Storm surge simulation along the Meghna estuarine area: an alternative approach
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Gour Chandra PAUL1, *, Sukumar SENTHILKUMAR2, Rana PRIA3
Affiliations
  • 1 Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh
  • 2 School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang-11800, Malaysia
  • 3 Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh
Published: 2018-01-25 doi: 10.1007/s13131-018-1157-9
Outline
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In this study, numerical prediction of surges associated with a storm was made through the method of lines (MOL) in coordination with the newly proposed RKARMS (4, 4) method for the meghna estuarine region, along the coast of Bangladesh. For this purpose, the vertically integrated shallow water equations (SWEs) in Cartesian coordinates were firstly transformed into ordinary differential equations (ODEs) of initial valued, which were then soloved using the new RKARMS (4, 4) method. Nested grid technique was employed for resolving the complexities of the region of interest with minimum cost. Fresh water discharge through the lower Meghna River was taken into account along the north east corner of the innermost child scheme. Numerical experiments were performed with the severe cyclone on April 1991 that crossed the coast over the study area. Simulated results by the study were found to be in good agreement with some reported data and were found to compare well with the results obtained by the MOL in addition with the classical 4th order Runge-Kutta (RK (4, 4)) method and the standard finite difference method (FDM).

method of lines  /  storm surge  /  nested grid  /  finite difference method  /  RKARMS (4, 4) method  /  truncation errors
Gour Chandra PAUL, Sukumar SENTHILKUMAR, Rana PRIA. Storm surge simulation along the Meghna estuarine area: an alternative approach[J]. Acta Oceanologica Sinica, 2018 , 37 (1) : 40 -49 . DOI: 10.1007/s13131-018-1157-9
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1 Introduction
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New tasks and paradigms are emerging in surge analysis because of the availability of information technology and increased amount of support data for building nonlinear system models. Accurate precipitation forecasts essential in the prediction of water levels due to surge with numerical models have proved to be useful for disaster planning, coastal management and reduction of losses resulting from storm surges (e.g., Das, 1972; Paul et al., 2014), and research is still being carried out towards the same for finding out more accurate solution. However, designing of a novel technique is always desirable in research community and its development. The method of lines (MOL) is a well-established numerical procedure for the treatment of boundary value problems (BVPs) providing a mathematical description of space and time. The method has no relative convergence problem and numerical instability (Sun et al., 1993), which can be used for saving computational cost in solving partial differential equations (PDEs) (Paul et al., 2014). The method is more effective over the finite difference method (FDM) with regard to accuracy and computational cost (Sadiku and Garcia, 2000). In simulating Indonesian tsunami 2004 along the coastal belt of Peninsular Malaysia and Thailand, Ismail et al. (2007) used the MOL in coordination with the RK (4, 4) method to solve vertically integrated shallow water equations (SWEs). They found the MOL to be an effective algorithm for solving SWEs, and to be a suitable procedure for the analysis of long waves originated in the shallow water. Keeping the idea in mind, Paul et al. (2014) solved SWEs for foreseeing water levels along the coast of Bangladesh associated with the severe cyclone in April 1991. They also found the MOL to be suitable technique over the FDM in solving SWEs in terms of accuracy, stability, and central processing unit (CPU) time. The basic idea of the MOL for solving time dependent PDEs is to approximate the spatial partial derivatives to obtain a system of ordinary differential equations (ODEs) of initial valued in most cases and thereby to solve applying any well-developed time integrators. In solving SWEs by the MOL, Paul et al. (2014) applied the RK (4, 4) method, where they concluded that the efficiencey of the MOL solution can be enhanced by using any well–established time integrators.
It is to be noted here that one of the limitations in implementing the RK (4, 4) method is the absence of error estimation procedure in estimating numerical solutions (Yaacob and Sanugi, 1998). Several methods have been developed to overcome this weakness. Pioneering works in this regard are due to Merson (1957) and Fehlberg (1969). Many details in the process since then have been advanced through the investigations of Yaakub and Evans (Evans and Yaakub, 1995; Yaakub and Evans, 1999; Yaacob and Sanugi, 1998; Ponalagusamy and Senthilkumar, 2009; Senthilkumar, 2009). Recently Ponalagusamy and Senthilkumar (2009) proposed a novel fourth order embedded RKARMS (4, 4) method to solve the real time application problems efficiently in image processing under cellular neural network (CNN) model and algorithm based on Runge-Kuttaa (RK) arithmetic root mean square with error control in detail. A detailed illustration related to local truncation error (LTE), global truncation error (GTE), and error estimation (EE) and control for the method can be found in Senthilkumar (2009). Paul and Senthilkumar (2016) tested the efficiencey of the RKARMS (4, 4) method in investigating protoplanetary structure, where they found the results by the method to be insensitive of the choice of the end points.
In this paper, we intend to simulate surge (in the absence of astronomical tide) associated with a strom along the Meghna estuarine region of the coast of Bangladesh by the MOL in coordination with the RKARMS (4, 4) method and to see how this different approach compare the results obtained with other two established methods, namely the FDM and the MOL in coordination with RK (4, 4) method and some reproted results.
The remainder of the paper is organized as follows. Section 2 deals with the model of SWEs with boundary conditions. A short note on the novel RKARMS (4, 4) method is addressed in Section 3. Numerical procedures are detailed in Section 4. Simulated results and analysis are presented in Section 5, and conclusions are given in Section 6.
2 Basic equations
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Prediction of water levels due to surge will be investigated based on the vertically integrated SWEs (Debsarma, 2009; Paul et al., 2014):
∂ζ ∂t =− ∂ ∂x [(ζ+h)u]− ∂ ∂y [(ζ+h)v],
∂u ∂t =−u ∂u ∂x −v ∂u ∂y +fv−g ∂ζ ∂x + T x ρ(ζ+h) − C f u ( u 2 + v 2 ) 1/2 ζ+h ,
∂v ∂t =−u ∂v ∂x −v ∂v ∂y −fu−g ∂ζ ∂y + T y ρ(ζ+h) − C f v ( u 2 + v 2 ) 1/2 ζ+h ,
where x and y are coordinate axes directed towards the south and the east, respectively; u and v represent Reynolds averaged components of velocity in the directions of x and y, respectively; ζ(x, y, t) is the displaced level of the surface of the sea; h(x, y) is the position of the sea bed; f=2Ωsin φ is the Coriolis parameter, where Ω(=7.29×10–5 rad/s) is the angular speed of the Earth rotation and φ is the latitude of the place, wherever necessary; g is the acceleration due to gravity; ρ(=1.03×103 kg/m3) is the density of the sea water, assumed to be homogeneous; Cf is the friction coefficient; (Tx, Ty) represents the components of wind stress.
In this study, surface stresses are parameterized by the conventional quadratic law:
( T x , T y )= ρ a C D ( u a 2 + v a 2 ) 1/2 ( u a , v a ),
where CD is the drag coefficient, ρa is the air density and ua and va are the x and y components of surface wind, respectively. The surface wind field over the model domain is derived following Paul et al. (2014) from the empirical formula due to Jelesnianski (1965), which is given by
V a ={ V 0 ( r a /R ) 3   for  all   r a ≤R, V 0 ( R/ r a )     for  all   r a >R,
where V0 represents the maximum sustained wind at the radial distance R from the eye of the cyclone and ra is any radial distance from the eye at which the wind field is desired. It is to be noted here that the parameters involved in Eq. (5), namely R and V0 are available during any storm along the region of interest at the Bangladesh Meteorological Department (BMD).
There are two types of boundaries in our model domain: closed and open boundaries. The normal component of the depth averaged velocity at the closed boundary (mainland and island boundaries) is set to zero and for open boundaries, a condition of radiation type is used to allow the disturbance, generated within the model area, to go out through the open boundary (Paul et al., 2016). Apart from the coast of Bangladesh, the boundaries are treated as straight lines in the open sea, which are given by (Roy, 1995)
west  boundary( 85°E ):  v+ ( g/h ) 1/2 ζ=0,
east   boundary( 95°E ) :     v− ( g/h ) 1/2 ζ=0 ,
south  boundary( 15°N ):  u− ( g h ) 1/2 ζ=0.
3 A short note on RKARMS (4, 4) method
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As in Paul and Senthilkumar (2016), the RKAM (4, 4) method can be written in the Butcher array form as
y n+1 = y n + h 3 ( k 1 + k 2 2 + k 2 + k 3 2 + k 3 + k 4 2 ),
where
k 1 =f( x n ,   y n ),
k 2 =f( x n + h 2 ,    y n + h k 1 2 ),
k 3 =f( x n + h 2 ,    y n + h k 2 2 ),
k 4 =f( x n +h,    y n +h k 3 ).
The fourth order RK root mean square (RKRMS (4, 4)) method with Butcher array can be written in the modified form as (Paul and Senthilkumar, 2016)
The RKRMS (4, 4) method due to Evans and Yaakub (1995) is given by
y n+1 = y n + h 3 [ k 1 2 + k 2 2 2 + k 2 2 + k 3 2 2 + k 3 2 + k 4 2 2 ],
where
k 1 =f( x n , y n ), k 2 =f( x n + 1 2 h, y n + 1 2 h k 1 ), k 3 =f( x n + 1 2 h, y n + 1 16 h k 1 + 7 16 h k 2 ), k 4 =f( x n +h, y n + 1 8 h k 1 − 17 56 h k 2 + 33 28 h k 3 ).
It is well-known that combination of RKAM (4, 4) and RKRMS (4, 4) (Eqs (9) and (10)) leads to give a new formation of RKARMS (4, 4), and is formulated by Ponalagusamy and Senthilkumar (2009) as
k 1 =f( x n , y n )= k 1 * , k 2 =f( x n + h 2 , y n + h k 1 2 )= k 2 * , k 3 =f( x n + h 2 , y n + h k 2 2 ), k 4 =f( x n +h, y n +h k 3 ), k 3 =f( x n + 1 2 h, y n + 1 16 h k 1 + 7 16 h k 2 )= k 3 * , k 4 =f( x n +h, y n + 1 8 h k 1 + 17 56 h k 2 + 33 28 h k 3 )= k 4 * ,
y n+1 = y n + h 3 [ k 1 + k 2 2 + k 2 + k 3 2 + k 3 + k 4 2 ],
y n+1 * = y n + h 3 [ k 1 *2 + k 2 *2 2 + k 2 *2 + k 3 *2 2 + k 3 *2 + k 4 *2 2 ].
As in Paul and Senthilkumar (2016), the embedded RKARMS (4, 4) method can be expressed in array form as
where
b T = y n+1 AM = y n + h 3 [ k 1 + k 2 2 + k 2 + k 3 2 + k 3 + k 4 2 ],
b ^ T = y n+1 RMS = y n + h 3 [ k      1 *2 + k 2 *2 2 + k 2 *2 + k 3 *2 2 + k 3 *2 + k 4 *2 2 ],
and the estimation of the LTE, E T =| b T − b ^ T |. In the RKARMS (4, 4) method, four stages are required to obtain the solution, which share the same set of vectors k1 and k2 using bT and b ^ T approximately, but k3 and k4 use bT while k 3 * and k 4 * use b ^ T . The derivation of the LTE and EE for the RKARMS (4, 4) method can be found in detail in Senthilkumar (2009) and Paul and Senthilkumar (2016).
4 Numerical procedures
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4.1 Set up of nested numerical schemes
The meghna estuary of Bangladesh is the world’s most vulnerable region because of the existence of thickly populated big and small islands, complex land-sea interface (Fig. 1), flat shallow bed, and shape of the coast (Roy, 1995; Roy et al., 1999). Besides, storm surges in the region of interest are amplified by some other factors found in Paul et al. (2016). An accurate prediction of surge levels for this region necessities proper understanding of the factors and their accurate incorporation. A use of high resolution of grids may be a solution in this regard (Paul and Ismail, 2012). However, with the use of the high resolution for the hole domain, as we will see later, computational cost may be increased tremendously, which is undesirable for a model of practical forecasting (Roy et al., 1999; Paul et al., 2014). The nested grid method is thought to be a probable solution in this regard (Paul and Ismail, 2012, 2013). The features of these schemes are that they allow specification of a desired resolution in a region of interest with lower resolution elsewhere (Paul et al., 2016). In our study, a high-resolution fine mesh numerical scheme (FMNS), capable of incorporating all major islands, is nested into a coarse mesh numerical scheme (CMNS). To take into account the complex land-sea interface and to incorporate bottom topographic details, a very high resolution numerical scheme (VFMNS) covering our region of concern is again nested into the FMNS. Some basic information on the schemes is given in Table 1. The coupling of our used schemes is made following the procedure addressed in Paul et al. (2016).
The Meghna River fresh water discharge is considered along the northern boundary of the VFMNS between 90.4°–90.6°E (Fig. 1) and the fresh water discharge of the river is incorporated according to Roy (1995), which is given by
u b =u+ Q B(h+ζ) ,
where Q and B are river discharge and breadth of the river along the longitude, respectively.
4.2 Discretization of spatial derivatives
The governing equations given by Eqs (1)–(3) as well as the boundary conditions given by Eqs (6)–(8) are discretized by finite-difference (central) for spatial derivatives only using the well-known Arakawa C-grid by considering the discrete points in the xy-plane defined by x i =(i−1)Δx, i=1,  2,  3,  ...,  M (even), y j =(j−1)Δx, j=1,  2,  3,  ...,  N (odd). Since the C-gid system is well-known and hence details of it are not presented here. However, it can be found in Paul and Ismail (2012).
Suppose that χ(x, y, t) represents any dependent variable at a grid point ( x i ,  y j ) at time tk. We define the following operators:
χ( x i ,  y j ,  t k )= χ i, j k ,  1 2 ( χ i+1, j k + χ i−1, j k )= χ i, j k ¯ x ,
1 2 ( χ i, j+1 k + χ i, j−1 k )= χ i, j k ¯ y , 1 4 ( χ i+1, j k + χ i−1, j k + χ i, j+1 k + χ i, j−1 k )= χ i, j k ¯ xy .
Then Eqs (1)–(3) can be written, respectively, as follows:
( ∂ζ ∂t ) i,j =CR1+CR2,
where i=2,  4,  ... , M−2 and j=3,  5,  ... ,  N−2.
( ∂u ∂t ) i,j =UR1+UR2+UR3+UR4+UR5+UR6,
where i=3,  5,  ... ,  M−1 and j=3,  5,  ... , N−2.
( ∂v ∂t ) i,j =VR1+VR2+VR3+VR4+VR5+VR6,
where i=2,  4,  ... , M−2 and j=2,  4,  ... , N−1.
In Eqs (16)–(18),
CR1=− ( ζ i+1,  j k ¯ x + h i+1, j ) u i+1, j k −( ζ i−1, j k ¯ x + h i−1, j ) u i−1, j k 2Δx ,
CR2=− ( ζ i, j+1 k ¯ y + h i, j+1 ) v i, j+1 k −( ζ i, j−1 k ¯ y + h i, j−1 ) v i, j−1 k 2Δy ,
UR1=−{ u i, j k u i+2, j k − u i−2, j k 4Δx ,                             for    i≠m−1 u i, j k 0.5(3 u i, j k − u i−2, j k )− u i−2, j k 4Δx ,   for   i=m−1
UR2 ​=​ − v i,⋯j k ¯ xy u i, j+1 k ¯ y − u i, j−1 k ¯ y 2Δy ,   UR3= f i, j v i, j k ¯ xy ,  UR4=−g ζ i+1, j k+1 − ζ i−1, j k+1 2Δx ,
UR5= T x ρ( ζ i, j k+1 ¯ x + h i, j ) ,  UR6=− C f u i,j k ζ i, j k+1 ¯ x + h i, j [ ( u i, j k ) 2 + ( v i, j k ¯ xy ) 2 ] 1/2 ,
VR1=−{ u i, j k ¯ xy v i+1, j k ¯ x − v i−1, j k ¯ x 2Δx ,                                       for    i≠2 u i, j k ¯ xy v i+1, j k ¯ x −0.5(3 v i, j k − v i+2, j k ) 2Δx ,               for   i=2
VR2=−{ v i, j k v i, j+2 k − v i, j−2 k 4Δy ,                           for    j≠2,  j≠n−1 v i, j k v i, j+2 k −0.5(3 v i, j k − v i, j+2 k ) 4Δy ,    for   j=2 v i, j k 0.5(3 v i, j k − v i, j−2 k )− v i, j−2 k 4Δy ,    for   j=n−1
VR3=− f i, j u i, j k ¯ xy ,  VR4=−g ζ i, j+1 k+1 − ζ i, j−1 k+1 2Δy ,  VR5= T y ρ( ζ i, j k+1 ¯ y + h i, j )
and
 VR6=− C f v i, j k ζ i, j k+1 ¯ y + h i,j [ ( u i, j k ¯ xy ) 2 + ( v i, j k ) 2 ] 1/2 .
From the boundary conditions given by Eqs (6)–(8), the elevations at j=1 (western), j=N (eastern) and i=M (southern) are computed, respectively, in the following manner.
ζ i, 1 k+1 =− ζ i, 3 k+1 −2 ( h i, 2 /g)    v i, 2 k  ,
ζ i, N k+1 =− ζ i, N−2 k+1 +2 ( h i, ⥄N−1 /g)    v i, N−1 k   ,
ζ M, j k+1 =− ζ M−2,⋯j k+1 +2 ( h M−1, j /g ) u M−1, j k .
where i=2, 4, 6, …, M–2 and j=1, 3, 5, 7, …, N.
The discretized equations presented above are also addressed in Paul et al. (2014), but due to subsequent relevant convenience, they are also presented in the study.
4.3 Data set up and initial conditions
The input data in our study are collected and compiled from various sources. The map of the study area was made with Google My Maps through ArcGIS software with which the three schemes, namely CMNS, FMNS, and VFMNS were created. The schemes were digitized according to their resolutions and were made usable for implementing our proposed method following an algorithm addressed in Murshed et al. (2016). The required meteorological inputs, namely maximum sustained wind speed, maximum sustained wind radius, and storm track (Fig. 2) were obtained from the BMD. It is to be mentioned here that the data related to the storm April 1991 were collected, as the numerical experiments which were performed with this storm. For specifying water depth data at the grid points of our used schemes, firstly a contour map for the region of interest is made from the British Admiralty Chart. Then from that contour map, an interpolated bathymetry is generated using Shepard interpolation with the help of a Matlab program. Contours of our interpolated bathymetry are delineated in Fig. 3. Water depth data at the grid points of the used three schemes are compiled using again the same interpolation with the use of that interpolated bathymetry. In our numerical calculations, we have used Cf=0.002 6, CD=0.002 8 (see Paul et al., 2016), Q=5 100.0 m3/s (Jain et al., 2007). The initial values of ζ, u and v are taken as zero to represent a cold start. An initial time step of 60 seconds is used that ensures Courant– Friedrichs–Lewy (CFL) stability criterion. The rest of the parameters involved in the problem have been assumed to have their standard values.
4.4 Solution by the novel RKARMS (4, 4) technique
For every grid point (xi, yj), one can see Eq. (16) to have the following form with CR1 and CR2:
( ∂ζ ∂t ) i, j = f 1 ( ζ l, m k ,   u l, m k ,   v l, m k ,   h l, m ),
where l=i−1,  i,  i+1;  m=j−1,  j,  j+1; i=2,  4,  ... , M−2 and j = 3, 5, ..., N – 2.
Again, for every grid point (xi, yj), where i=3, 5, …, M − 1 and j=3, 5, …, N − 2, Eq. (17) can be written as
( ∂u ∂t ) i, j = f 2 ( ζ l, m k+1 , u l, m k , v l, m k , h l, m ),
where l=i−2,  i−1,  i,  i+1,  i+2  and  m=j−2,  j−1,  j,  j+1,  j+2.
Finally, for every grid point (xy, yj), Eq. (18) can be written as
( ∂v ∂t ) i, j = f 3 ( ζ l, m k+1 ,  u l, m k ,  v l, m k ,  h l, m ),
where l = i − 2, i − 1, i, i + 1, i + 2; m = j − 2, j − 1, j, j + 1, j + 2; i = 2, 4, ..., M − 2 and j = 2, 4, ..., N − 1.
In Eqs (22)–(24), the superscripts k and k+1 indicate the values of the variable at kth and (k+1)th time levels, respectively. The known values of ζ l, m k ζ l, m k , ζ l, m k+1 , u l, m k , v l, m k and h l, m on the right hand side of Eqs (22)–(24) are used as the initial conditions for integrating the corresponding equation at every time instant. Firstly, Eq. (22) with the initial conditions of the variables involved (discussed in Subsection 4.3) is solved with the help of the novel RKARMS (4, 4) method to update the values of ζ at the interior points, Eqs (19)–(21) are then used to upgrade the values of ζ at the boundary points. Finally, average procedure is adopted to get elevations at the remaining grid points representing water and coastal or island boundaries. Track of the cyclone is then generated with the obtained data from the BMD, associated wind field is generated subsequently using Eqs (4) and (5). Equation (23) is then integrated by the novel RKARMS (4, 4) technique and finally Eq. (24) is solved. The procedure is repeated over time updating the values of the variables ζ, u, and v at each time step of the solution process. The same procedure is employed to all the schemes, where the only difference is in the boundaries. There are three open boundaries of the CMNS, where a condition of radiation type given by Eqs (19)–(21) is applied. The results along three particular grid lines of the CMNS are coupled with those of the open boundaries of the FMNS. The grid lines of the CMNS which are adjacent to those of the FMNS are chosen. The coupling of the schemes is made by a weighted interpolation. A similar manner is adopted in coupling the FMNS and VFMNS. Along the northern side of the VFMNS, ub is computed at points (1, j) using Eq. (15) in which u on the RHS is replaced by u3, j, where j = 7, 9, 11, 13, 15, 17, 19.
5 Results and discussion
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It is aforementioned that our numerical experiments were carried out with the April 1991 cyclone as it crossed the coast over the analysis area. For analysis of results, the time history of the storm is of help. The details of it are addressed in Paul et al. (2016), and hence it is not presented in this study. But for convenience, the track of the storm and the corresponding table are presented here (Fig. 2 and Table 2), where the data were received from the BMD. The results were calculated for 72 h and the water levels at some grid points representing representative stations were presented for the last 48 h for better perspective.
Figures 4 and 5 depict our model simulated surge levels associated with the cyclone April 1991 at some locations in the analysis area. It can be seen from the figure that our simulated maximum surge levels in this area vary between 5.16–7.03 m and the surge level at Chittagong is found to be 5.39. Rahman et al. (2013) estimated 4.04–6.01 m surge along the Meghna estuarine region, Paul et al. (2014) predicted 5–7 m high surge in this area, Paul et al. (2016) estimated a storm surge of 6–8 m along the estuary of Meghna, whereas Dube et al. (2004) computed a peak surge value of 5.8 m at Chittagong. By a report from the BMD, the maximum surge level at Chittagong was 5.5 m and there was 3.5–6.1 m surge along the coast of Bangladesh (Paul et al., 2014). Thus our computed peak surge values in our region of interest are in a reasonable agreement with those obtained in different investigations and the peak surge value at Chittagong compares well with the reported surge height by the BMD. According to a report from the BMD, the cyclone made landfall near the north of Chittagong coast (at Companigonj) about 20: 00 UTC on April 29. Our simulated results presented in Fig. 4 justify the landfall time of the storm. As the astronomical tide is not considered here in the study, so the estimated land fall time may vary. It is pertinent to point out here that astronomical tides cannot be found switched off in the nature and therefore one cannot observe a surge and hence comparison of our time variation of surge levels with observation is not done. Surge levels (Figs 4 and 5) around Chittagong were found to be higher than that of the levels at other locations of the estuary. This is as to be expected as the storm crossed the coast at Companigonj, which is just north of Chittagong (Paul and Ismail, 2013) and may be this is the reason of highest surge value (7.03 m) at Companigonj (Figs 2 and 4a), which is supported by Flather (1994). It can be seen from Fig. 4 that the maximum surge level at each location increase with time as the storm approaches the coast and finally there is a recession. This is expected as the circulatory wind attains highest intensity during crossing the coast. A similar analysis can be found to make in some investigations, namely Paul and Ismail (2012, 2013) and Paul et al. (2014, 2016). The beginning of recession at the coastal locations delays with the increase in their east longitudes (Fig. 4). The track of the cyclone may be one of the reasons behind the fact.
The study was also conducted employing the FDM (forward in time and central in space) as well as employing the MOL in coordination with the RK (4, 4) method using the same nested schemes, same grid, same coupling system of the schemes, same set of equations with same boundary conditions and same values for the parameters that were used in the case of the present study. Figure 5 provides a comparison of time variation of water levels due to surge simulated by the present study and the other two methods of interest at Char Chenga, Chittagong and Sandwip. Figure 5 shows that surge levels simulated by our model compare well with the ones computed by the FDM and by employing the MOL in coordination with the RK (4, 4) method. Surge levels computed by our model for other locations of the estuary also compared well with the ones computed by the said two methods. Other such figures are excluded due to the sake of brevity.
To test the effect of Meghna River fresh water discharge, grid resolution, Coriolis force and offshore islands on surge levels, the model was run without including each of the factors namely, the river discharge, VFMNS scheme, Coriolis force and offshore islands keeping all other factors fixed. The results for the peak surge values are presented in Fig. 6. It can be seen from the figure that the region’s surge levels are modified by the offshore islands and Coriolis force. It can also be inferred from the figure that surge levels increase at each of the stations when discharge is taken into account and the resolution of grids can be a factor for foreseeing surge levels accurately in the area of interest. To investigate the contribution of water depth on surge levels, the model is run with a constant water depth of 10 m. The peak surge values in this regard are also presented in Fig. 6. In that case water levels were found to be amplified. Thus for proper estimation of surge levels, water depth should properly be taken into account.
In our computation, initial time step size, Δt, is used as 60 s. Although, we have presented our results taking time step Δt as 60 s, we have also computed our results taking different time steps, namely 90 s, 30 s, 20 s, and 10 s. The results obtained employing the MOL in coordination with both the RKARMS (4, 4) and RK (4, 4) methods in each case generally agreed the results presented in each case and no numerical instability was achieved. But the FDM failed to ensure stability for some time steps mentioned above. In order to compare the computational efficiency of the methods, the codes were run on the same computer with initial time step 60 s. The computational time in the case of the present study was found to be a little bit more in comparison with the other two, but with the use of higher time step it can be reduced, as no numerical instability was obtained with higher time step in the case of the present study and there is a facility of controlling step size automatically based on the LTE by the method.
Further development is yet to be made with respect to its refinement and inclusion of astronomical tide. It is pertinent to pinpoint out here that tide can be generated by the cubic spline interpolation with the available hourly tidal data from the BIWTA and that can be added with time variation pure surge values obtained in the study to obtain the total water levels. Since our main intention was to solve SWEs by the new different approach adopted in this study, astronomical tide was not incorporated.
6 Conclusions
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Ensuring robustness of the solutions is essential through new RKARMS (4, 4) method if the significant practical benefits of vertically integrated SWEs in Cartesian coordinates are solved using the MOL to predict water levels due to surge along the world’s most vulnerable region, the Meghna Estuary. The region’s surge levels are found to be influenced by some factors, namely river discharge, grid resolution, offshore islands, water depth and Coriolis force. Thus for prediction of surge levels with a considerable accuracy along the region of interest, one needs to incorporate the above factors properly in a model with a precise understanding of the factors. However, simulated surge levels by the study are found to be in good agreement with some reported data and are found to compare well in comparison with the results obtained by the FDM and the MOL in coordination with the RK (4, 4) method. Indeed, the MOL approach minimizes computational effort and cost. It has the advantages of computational efficiency, numerical stability and a significant benefit of using sophisticated ODE integrator. On the other hand, RKARMS (4, 4) can control step size based on the LTE, which in turn can reduce computational cost. Hence the method adopted in the study can be an alternative approach for the prediction of surge levels associated with a storm along the coast of Bangladesh.
Acknowledgements
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The authors are thankful to the anonymous reviewers for their insightful comments in improving the quality of the manuscript. The first author would like to thank Md. Mizanur Rahman, from Department of Mathematics, Shahjalal University of Science & Technology, Sylhet, Bangladesh for providing necessary data and Figures. The second author would like to extend his sincere gratitude to Professor Ramasamy Ponalagusamy, Department of Mathematics, National Institute of Technology [REC], Tiruchirappalli-620015, Tamilnadu, India for his valuable guidance during doctoral research work. Rana Pria is an M.Sc. student of Mathematics Discipline, Khulna University, Bangladesh, who would like to thank her supervisor for his valuable suggestions.
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Year 2018 volume 37 Issue 1
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Article Info
doi: 10.1007/s13131-018-1157-9
  • Receive Date：2016-11-05
  • Online Date：2026-04-13
  • Published：2018-01-25
Article Data
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  • Received：2016-11-05
  • Accepted：2017-06-22
Affiliations
    1 Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh
    2 School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang-11800, Malaysia
    3 Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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