Comparison of Numerical Method for Forward. and Backward Time Centered Space for. Long - Term Simulation of Shoreline Evolution

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1 Applied Mathematical Sciences, Vol. 7, 13, no. 14, HIKARI Ltd, Comparison of Numerical Method for Forward and Backward Time Centered Space for Long - Term Simulation of Shoreline Evolution Suiyanto 1, Mohammad Fadhli Ahmad 1, Mustafa Mamat, and Mohd Lokman Husain 3 1 Department of Maritime Technology Department of Mathematics 3 Institute of Oceanography and Environment, Universiti Malaysia Terengganu, 13 Kuala Terengganu, Terengganu, Malaysia suiyanto17@gmail.com, fadhli@umt.edu.my Copyright 13 Suiyanto et al. This is an open access article distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. Astract Mathematical modeling of shoreline evolution ecomes a useful engineering technique for investigating and predicting the evolution of the plan view of the sandy each. In this paper two numerical schemes for the shoreline evolution in the long-term scale are presented, and comparisons to analytical solution for some cases are presented for a satisfactory level of suitale numerical scheme. Analytical solutions of shoreline evolution for simple configuration are presented under idealized wave condition. Based on the result that ackward time centered space scheme more suitale than forward time centered space scheme to simulate shoreline evolution in the long-term scale. Mathematics Suect Classification: 6L1; 93A3; 37M Keyword:mathematical modeling, shoreline evolution, long-term scale, ackward time centered space, forward time centered space

2 166 Suiyanto et al. Introduction Three time scales of shoreline evolution which can e distinguished are geological evolution over hundreds and thousands of years, long-term evolution from year to year or decades and short-term or seasonal evolution during a maor storm. For the prolem under consideration, long-term evolution is the primary importance ecause the long-term evolution is water wave or wave-generated currents. Three phenomena intervene in the action which wave has on shoreline evolution are erosion of each material y short period seas versus accretion y longer period swells, effect of water level changes on erosion and effect of coastal structures [1]. In order to investigate of each ehavior is needed qualitative understanding of idealized shoreline response to the governing process. Analytical solution originating from a mathematical model which descries the asic physics is the one tool to understanding it. The analytical solutions are often valuale for giving qualitative insight and understanding the properties of shoreline change in the long-term scale. The analytical solution cannot e expected to provide quantitatively accurate solutions to prolems involving complex oundary conditions and wave inputs. In the real situation, a numerical model of shoreline evolution would e more appropriate. Many authors otained an analytical solution to shoreline evolution y using a simple mathematical formula. The one-line theory was introduced y many authors, several contriutors in the analytical solution of shoreline evolution includegrim[, 6], Bakker and Edelman [3], Bakker [4], Le Mahute and Soldate [9], Walton and Chiu [13], and Larson et al. [8]. Two numerical schemes of shoreline evolution for simplified configuration each are examined and presented in this paper. Fundamental assumptions of the model In the one-line model, the each profile is assumed to move landward and seaward while retaining the same shape, implying that all ottom contours are parallel. Consequently, under this assumption it is sufficient to specify the horizontal location of the profile with respect to aseline, and one contour line can e used to descrie changes in the each plan shape and volume as the each erodes and accretes. The maor assumption of the model is the sand is transported alongshore etween two well-defined limiting elevations on the profile. One contriution to the volume change results if there is a difference in the alongshore sand transport rateat the lateral sides of the section and the associated the sand continuity. The principles of mass conservation must apply to the system at all times. By considering aove definitions, the following differential equation for shoreline evolution is otained: y = t ( D + D ) B 1 C Q x (1)

3 Comparison of numerical method 167 wherex is the alongshore coordinate (m); y is the shoreline positions (m) and perpendicular to x-axis; t is time (s); Qis the long-shore sand transport rate (m 3 /s); DB isthe average erm height (m)and D C is the closure depth(m). In order to solve the equation (1), necessary to specify an expression for the longshore sand transport rate, Q. This quantity is considered to e generated y the wave oliquely incident to the shoreline. A general expression for the long-shore sand transport rate was developed y the US Army Corp [1]: Q = Q sin α () ( ) where Q is the amplitude of the long-shore sand transport rate. The empirical predictive formula for the amplitude of the long-shore sand transport rate is[7]: ρ K Q = ( H cg ) (3) 16 ( ρ s ρ )( 1 n) where the suscript denotes value at the point of reaking, c g is the wave group velocity, H is the wave height, ρ s is the density of the sediment (kg/m 3 ), ρ is the density of the sea water, n is the porosity and K is the dimensionless coefficient which is a function of particle size. The quantity α is the angle etween reaking wave crest and local shoreline, and may e written as: 1 y α = α tan (4) x where α is the angle etween reaking wave crests and the x-axis. For eaches with mild slope, it can e assumed that reaking wave angle to the shoreline is small. In this case, sin( α ) α and tan -1 y. Sustituting x equation (4) into the equation(), and assuming the each with mild slope, yields: y Q = Q α () x Sustituting equation () into the equation (1) and neglecting the sources or sinks along the coast gives: y y = D (6) t x Q where D =. Equation (6) is analogous to the one-dimensional heat D B + D C diffusion equation, it can e solved analytically for various initial and oundary conditions. Numerical Scheme For practical prolem, the equation of shoreline evolution and oundary condition cannot normally e simplified sufficiently for the analytical solution to e valid. In that case, the equation of shoreline evolution must e solved numerically.

4 168 Suiyanto et al. Numerical schemes that use in this paper are Forward Time Centered Space (FTCS) and Backward Time Centered Space (BTCS). FTCS is the numerical scheme uses finite difference technique and is stepped forward in time using increments of time interval [1, 14]. The information used in forming the finite difference quotient in FTCS comes from aove of grid point ( i, ); that is, it uses y i, + 1 as well as y i,. No information to the ottom of ( i, ) is used (see Figure1a). While BTCS is the numerical scheme uses finite difference technique and is stepped ackward in time using increments of time interval [1, ]. The information used in forming the finite difference quotient in FTCS comes from ottom of grid point ( i, ); that is, it uses y i, 1 as well as yi,. No information for the aove of ( i, ) is used (see Figure 1). Both of the FTCS and BTCS use the finite difference quotient of space comes from oth sides of the grid point located at ( i, ); that is, it uses y i, + 1 as well as. i, falls etween the two adacent grid points [1]. y Grid point ( ) i, 1 (a) () Figure 1. Computational molecules: (a).ftcs scheme, (). BTCS scheme Finite difference expression for the FTCS equation (6)can e written as: t y i, + 1 = yi, + D ( yi 1, yi, yi 1, ) + + (7) x Finite difference expression for the BTCS equation (6)can e written as: t y i, D ( y 1,, 1, ) i+ yi + yi = yi, 1 (8) x To calculate the solution of y is needed initial and oundary condition. The initial condition is the value at all grid points at time level. For FTCS scheme to otain values y at time level + 1 are calculated from the known value at the time level. when these calculations are finished, the values y at time level + 1 are otained. This calculation will repeat until the final time level of simulation. While For BTCS scheme, the initial condition is the value at all grid points at time level 1. Equation (8) represents one equation with three unknown values y at

5 Comparison of numerical method 169 time level, namely, y i + 1,, i y, and y. Hence, equation (8) applied at a given grid point i does not stand alone; it cannot y itself result in a solution y i,. Rather equation (8) must e written at all interior grid points, resulting in system of algeraic equation from which unknowns y, for all i can e solved simultaneously. This method is usually involved with the manipulation algera of large matrices. i 1, i Result and discussion In order to investigate the shoreline evolution in the long-term scale.the numerical results of the different each situation are considered and the solution the idealized prolem is presented. During all these simulations, the value of the constant D =. m/year. Parameter. D depends on the wave climate and each material and has the role of diffusion coefficient [11]. Prolem 1. Straight Impermeale groin For this prolem, the initials of each is parallel to the x-axis with the same reaking wave angle ( α ) existing everywhere (see Figure ), thus leading to uniform sand transport rate along the each. At time t = a thin groin is instantaneously place at x = and lock all transport. Mathematically, this oundary condition can e formulated as follows[8]: y = tanα at x = (9) x This equation states that the shoreline at the groin is instant parallel to the wave crests. A groin interrupts the transport of the sand alongshore, cause an accumulation on the up-drift side and erosion on the down-drift side Groin Initial Shoreline Figure. Initial Shoreline with configuration straight impermeale groin. The analytical solution descriing the accumulation part on up-drift side of the groin is: x 4Dt x π x 4Dt y( x, t) = tanα e erfc (1) π Dt Dt

6 17 Suiyanto et al. In this case, α was set to., since the oundary condition at the groin which is totally locking the transport of sand alongshore so that long-shore sediment transport rate taken to e Q =. The shoreline changes calculated y using the analytical solution equation (1) and numerical solutions y using FTCS scheme and BTCS scheme are shown in Figure 3. The comparison etween the analytical and numerical solutions is only investigated on the up-drift side, since the analytical solution on the down-drift side has not een considered in this paper. In case of time durations are equal to 1 year and years, the FTCS scheme and BTCS scheme produce an almost identical shoreline to the analytical (see Figure 3a and Figure 3). This implies that these numerical scheme can handle these situations. When the time duration is increased to e years, the FTCS scheme cannot handle the solution. It is ecoming unstale. The solution ecomes unstale and periodic (see Figure 3c). While the BTCS scheme still can handle simulation until 1 years (see Figure 3c and Figure 3d). 16 time duration = 1 year 4 time duration = years (a) () 7 time duration = years 16 time duration = years, years and 1 years (c) (d) Figure 3. Shoreline evolution of interruption from straight impermeale groin. Prolem. Rectangular cut in each Thisprolem represents an excavation or natural employment of rectangular shape. The initial conditions for rectangular cut in a each are formulated as[13]:,, (11), The analytical solution is y a x a + x y( x, t) = erfc + erfc (1) Dt Dt

7 Comparison of numerical method 171 for t > and < x <. This situation is the inverse prolem of the rectangular each fill [8], so that this situation can e used to evaluate the rate of infilling of certain volumetric percentage of sand. In this prolem, y was set to 3 m and a was set to m, since this prolem doesn t lock the transport of sand alongshore so that the oundary condition is specified on the initial and final x-axis that depend from Eq. 1 (see Figure 4) Initial Shoreline Figure 4. Initial Shoreline with configuration a rectangular cut in an infinite each. The FTCS scheme and BTCS scheme produce an almost identical shoreline to the analytical during simulated until 4 years (see Figure a and Figure ). When the time duration is increased to e years, the FTCS scheme cannot handle the solution. It is ecoming unstale and grow lead into oscillation (see Figure c). While the BTCS scheme still can handle simulation until 1 years (see Figure c and Figure d). 3 time duration = 1 year 3 time duration = years (a) () 3 time duration = years 3 time duration = years, years and 1 years (c) (d) Figure. Shoreline evolution of a rectangular cut in an infinite each.

8 17 Suiyanto et al. Conclusion The comparison etween the analytical solution and numerical solution for two different shoreline situations under idealized wave condition are discussed. The otained result for the first case show that the FTCS and the BTCS scheme can handle numerical solution for straight impermeale groin fill finely until 14 years. When the time duration increased for the FTCS scheme cannot handle numerical solution for this prolem, while the BTCS scheme still can handle numerical solution until long-term duration. The otained result for the second case shows that the FTCS and the BTCS scheme can handle numerical solution for rectangular cut in each finely until 14 years. Similarly the first prolem, when time duration increased for the FTCS scheme cannot handle numerical solution for this prolem. While the BTCS scheme still can handle numerical solution until long-term duration. These resultsimply that the BTCS scheme more suitale than FTCS to simulate shoreline evolution in the long-term scale. Acknowledgement We would like to thank the financial support from Department of Higher Education, Ministry of Higher Education Malaysia through the Exploratory Research Grants Scheme (ERGS) Vot.79. References [1] J. D. Anderson, Computational Fluid Dynamics: The Basic with Applications, McGraw-Hill, Singapore, 199. [] W. F. Ames,Numerical Methods for Partial Differential Equation, Ed. nd, Academic Press, Inc., Florida, [3] W. T. Bakker and T. Edelman,. The Coastline of River Deltas. Proceeding of 9 th Coastal Engineering Conference, America Society of Civil Engineers,(196), [4] W. T.Bakker, The Dynamics of Coast with a Groin System. Proceeding of 11 th Coastal Engineering Conference, America Society of Civil Engineers, (1969), [] W. Grim,Theoretical Form of Shoreline, Proceeding of 7 th Coastal Engineering Conference, America Society of Civil Engineers,(1961), [6] W.Grim, Theoretical Form of Shoreline, Proceeding of 9 th Coastal Engineering Conference, America Society of Civil Engineers, (196), 19-3.

9 Comparison of numerical method 173 [7] L. X. Hoan, Some Result of Comparison etween Numerical and Analytical Solutions of the One-line Model for Shoreline Change. Vietnam Journal of Mechanics. 8() (6), [8] M.Larson, H. Hanson and N. C. Kraus, of the One-line Model for Shoreline Change, Technical Report CERC 87-, US Army Corps of Engineer Waterways Experiment Station, CERC,1987. [9] B. Le Mahute and M. Soldate, Mathematical Modeling of Shoreline Evolution. CERC Miscellaneous Report No 77-1, US US Army Corps of Engineer Waterways Experiment Station, CERC,1977. [1] B. Le Mahute and M. Soldate, A Numerical Model for Predicting Shoreline Changes. Miscellaneous Report No.8-6, US Army Corps of Engineering, Fort Belvior, CERC, 198. [11] D. Reeve, A. Chadwick and C. Fleming, Coastal Engineering: Processes, theory and design practice Ed. nd, Spon Press, New York,1. [1] US Army Corp of Engineers.Shore Protection Manual, Coastal Engineering Research Centre, Washington,1984. [13] T. Waltonand, T. Chiu, A Review of Analytical Technique to Solve the Sand Transport Equation and Some Simplified Solution, Proceeding of Coastal Structure, America Society of Civil Engineers,(1979), [14] W. Y.Yang, W.Cao, T. Chung, and J. Morris, Applied Method using Matla, A John Wiley & Sons, Inc., Canada,. Received: July, 13