Efficient numerical scheme based on the method of lines for the shallow water equations

Mohamed M. Mousa

a Department of Basic Science, Faculty of Engineering at Benha, Benha University, Benha 13512, Egypt

b Department of Mathematics, College of Sciences and Human Studies at Howtat Sudair, Majmaah University, 11952, Saudi Arabia

Abstract In this paper, a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines. The simulation is carried out for various coastal slopes and the ocean depths. The effects of the coast slope and sea depth on the tsunami wave run-up height and velocity are illustrated. The accuracy of the mathematical model is verified by solving a classical test problem with known analytic solution. The computed run-up height and velocity show satisfactory agreement with the tsunami wave physics.

Keywords: Shallow-water equation; Tsunami wave; Run-up height; Method of lines.

1.Introduction

A Tsunami is a wave train, or series of waves, generated in a body of water by an impulsive disturbance that vertically displaces the water column. Earthquakes, landslides, volcanic eruptions, explosions, and even the impact of cosmic bodies,such as meteorites, detonation of nuclear devices near the sea can give rise to such destructive sea waves so called tsunamis[1-3] . By far the most destructive tsunamis are generated from large shallow-focus earthquakes with an epicenter or fault line near or in the ocean. Vertical displacements of the earth's crust along the rupture resulting from such earthquakes can generate destructive tsunami waves which can travel across an ocean spreading destruction across their path as shown in Fig. 1 .

Fig. 1. Formation of tsunami waves.

Although the sources for formation of tsunami are considered as point sources, the tsunami waves generated can be very destructive locally, the energy of the waves is rapidly dissipated as they travel across the ocean, can ravage coastlines, causing property damage and loss of life. The speed of the tsunami is governed by the water depth. Speed reduces and wave height increases as it approaches the shore.Tsunamis have three stages: formation, mid-ocean propagation, and breaking and run-up on the beach. The last two stages will be discussed along this paper. One of the most important questions in tsunami modeling is the estimation of tsunami run-up heights at different points along a coastline.In order to determine run-up of long waves, different theoretical and experimental studies have been performed. Some of them are referenced in this section. The early experimental work reported by Hall and Watts [4] and Camfield and Street[5] have been used in the past to verify analytical results and the accuracy of numerical models (Li and Raichlen [6] ).Kobayashi et al. [7] developed a numerical flow model to predict the flow characteristics on rough slopes for specified,normally incident wave trains. The finite amplitude shallowwater equations including the effects of bottom friction are solved numerically in the time domain using an explicit dissipative Lax-Wendroff finite difference method. The effects of permeability are assumed to be negligible, so that the flow computation may be limited with the region on a rough slope.Kobayashi et al. [8] investigated wave run-up and reflection on a 1:3 rough impermeable slope for irregular wave. Kanoglu and Synolakis [9] studied long-wave evolution and run-up on piecewise linear one and 2D bathymetries analytically and experimentally with the objective of understanding certain coastal effects of tidal waves. They compared analytical predictions with numerical results, with results from a new set of Revere Beach and also with the data on wave run-up around an idealized conical island. Maiti and Sen [10] described a numerical time-simulation algorithm for analyzing highly nonlinear solitary waves interacting with plane, gentle and steep slopes by employing a mixed Eulerian-Lagrangian method. It is found that the run-up height is crucially dependent on the wave steepness and the slope of the plane.Pressures and forces exerted on impermeable walls of different slopes by progressive shallow-water solitary waves are studied. Li and Raichlen [6] , deal with the run-up of solitary waves on a uniform plane beach connected to an open ocean of constant depth. The waves are nonbreaking during the runup process. A nonlinear solution to the classical shallow-water equation, that describes the wave characteristics on the beach,is obtained analytically by using a hodograph transformation.Gedik et al. [11] carried out a laboratory investigation a tsunami run-up and erosion area on permeable slope beaches in a channel. They formulated a relation between run-up height and erosion area. Recently, Chao and Yongen [12] used a finite-element procedure that includes the interaction between solid and fluid based on the potential flow theory to simulate the dynamics of tsunami wave induced by a thrust fault earthquake in order to investigate the effect of different beach slopes on the tsunami run-up. They concluded that the source wave of a tsunami induced by a thrust fault earthquake is located above the hanging wall and then it splits into two tsunami waves -one above the hanging wall and the other above the foot wall-with different amplitudes, waveforms and velocities.

In this paper, the classical nonlinear shallow-water equations are considers as a model of tsunami run-up. Two cases are considered. The first case is the classical test problem with known analytical solution for which the beach slope is zero i.e. for a uniform sea depth. The second case considered is at which the beach slope is variable. Three beach slopes are considered in the second case. The effects of increasing of beach slope and the sea depth on the height and velocity of tsunami wave have been studied using the method of lines(MOL).

Fig. 2. Tsunami solitary wave run-up near a coast.

2.Shallow-water Tsunami model

Consider the specific case of the run-up of 2D long wave's incident upon a uniform sloping beach connected to an open ocean of uniform depth shown in Fig. 2 .

The classical nonlinear shallow-water equations are[11,13] :

whereu(x,t) is tsunami velocity, η(x,t) is the surface elevation (wave height),H(x) is the variable sea depth near the coast andgis the acceleration due to gravity (g= 9 . 8 m/ s2) .The initial conditions can be written as

wherehis an initial wave height anddis the sea depth in the open ocean. This problem has an exact solutions for the tsunami velocity and wave amplitude only when sea depth is constant,H(x) =d, i.e. mid-ocean propagation stage of tsunami. The exact soliton solution in this stage is defined in[11] as

The system consisting of Eqs. (1) -(4) has no exact solution whenH(x) is variable. So, we will solve this system for three different linear functionsH(x) in the formH(x) =mx+ 300,wherem= 0.2, 0.4 and 0.6, i.e. breaking and run-up on the beach stage of tsunami, in order to show the effect of the coast slopemon the run-up heights and the velocity of tsunami wave.

3.Method of lines formulation

Method of lines is a semi-discrete approach [14-17] which involves reducing initial/boundary value problems containing partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) in time through the use of a discretization in space. The most important advantage of the MOL approach is that it has not only the simplicity of the explicit methods but also the superiority (stability advantage) of the implicit ones unless a poor numerical method for solution of ODEs is employed. It is possible to achieve higher-order approximations in the discretization of spatial derivatives without significant increasing in the computational complexity. The MOL has a wide applicability to physical and chemical systems modeled by PDEs.

According to the MOL [14-17] , the coordinatexin Eqs. (1) -(4) is discretized withNuniformly spaced grid pointsxi=xi-1 + ?x,x0 =a,xN=b,i= 1 , 2, . . . ,N. Note that interval the [a,b] is the considered solution domain alongx.Here, ?x= (b-a) /N, so we can writexi=a+i?x. A second-order central difference scheme is used to approximate the first derivatives according to the spatial variablexat the grid pointsxi,i= 1 , 2, . . . ,N-1 . Discretizing using the second-order central difference and consideringui(t) and ηi(t) approximateu(xi,t) and η(xi,t), respectively, lead to the following system of the ODEs

in case of constant sea depth near the coast,H(x) =d, i.e.mid-ocean propagation stage, or

in case of variable sea depth,H(x) =mx+ 300, i.e. breaking and run-up on the beach stage. Here,i= 1 , 2, . . . ,N-1 ,t≥0, and the initial/boundary conditions of the obtained ODEs systems can be written as

Solving the ODEs systems corresponding to the initial/boundary conditions using the classical fourth order Runge-Kutta scheme (RK4) with a suitable time step ?t,one can obtain the solution of the considered problem at every grid point of the computational domain. We have used the algorithm of the classical RK4 built-in the Maple 12 package to solve the obtained ODE systems. All the cases considered in this work are solved in the computational domain of[ -400, 400] ?[0, 7] usingN= 1600, i.e. ?x= 0. 5 and the time step ?tis chosen to be 0.01. The initial wave heighthis considered to beh= 2.

4.Numerical results and discussion

We simulate the tsunami propagation along the coastline with the main purpose of validating the method of lines for solving shallow water equations. The simulation is done for two main cases: (1) mid-ocean tsunami propagation and (2)breaking and run-up of tsunami waves on the beach for 3 different coast slopes. The simulation is presented graphically in the positive portion of computational domain to further clarify the results after entering coast region.

4.1. Case 1: Mid-ocean tsunami propagation when H (x) = d

Results from the numerical simulation of the tsunami wave height and velocity in the open ocean are compared with the exact solution and shown in Fig. 3 to validate results accuracy of the MOL.

From Fig. 3 , it can be seen that the tsunami wave velocity and height maintain their shapes while it propagates at a constant velocity and height. This solitary behave of the tsunami wave is caused by a cancellation of nonlinear and dispersive effects in the medium. Here, the tsunami wave maintains its shape because there is no coast allows wave breaking. The effect of the ocean depth on wave is disused later after comparing with the case of breaking and run-up of tsunami waves on the beach. Also, Fig. 3 shows a satisfactory agreement between the MOL numerical results and the exact solution. This satisfactory can be increased by reducing ?x or ?t .

4.2. Case 2: Breaking and run-up of tsunami waves on the beach

In order to illustrate the breaking behavior of the tsunami wave on the beach, we have simulated the wave at three different sea depths, i.e.H(x) =mx+ 300, wherem= 0.2,0.4 and 0.6. The obtained results are presented graphically in Figs. 4 -7 . The tsunami wave velocity and height behaviors at different times and coast slopes are illustrated in Figs. 4 -6 .

The important observation that can be drawn from Figs. 4 -6 is that when the tsunami enters the shoaling water of coastlines in its path, the velocity of its waves diminishes and the wave height increases in the first moments and then decreases over time until it reaches the sea level. In other words, as a tsunami leaves the deep water of the open sea and propagates into the more shallow waters near the coast, it undergoes a transformation. Since the velocity of the tsunami is related to the water depth, as the depth of the water decreases, the velocity of the tsunami diminishes. The change of total energy of the tsunami remains constant. Therefore,the velocity of the tsunami decreases as it enters shallower water, and the height of the wave grows for a certain period and then decays over time until it reaches the sea level after the full breaking. Because of this “shoaling”effect, a tsunami that was imperceptible in deep water may grow to be several feet or more in height.

Fig. 3. A comparison of the MOL results (dotted line) and exact solution (solid line) at H(x) = d = 20 and t = 0, 2, 4 and 6 for the tsunami wave velocity and height.

Fig. 4. Tsunami wave velocity and height at d = 20, m = 0.2 and various times.

In order to illustrate the effect of the coast slope on the tsunami wave run-up height and velocity, we have plotted the MOL results of the wave height and velocity at three different coast slopes. The obtained results are presented graphically in Fig. 7 .

Fig. 5. Tsunami wave velocity and height at d = 20, m = 0.4 and various times.

Fig. 6. Tsunami wave velocity and height at d = 20, m = 0.6 and various times.

From Fig. 7 , it can be observed that the slope of the coast has the same effect on the wave velocity and height. As the slope of the coast decreases, the velocity and height of the tsunami wave increase. Therefore, a danger of the tsunami waves in case a beach of high slope is lower than in the case of slight slope beach. The effects of the ocean depthdon the tsunami wave height in deep water (open ocean) and near the coast are shown in Fig. 8 . In open ocean, the ocean depth affects only on the wave width while it has no effect on the wave height. If the ocean depth increases, the tsunami wave width increases as well while the wave height remain constant. However, the depthdhas effects on both of wave width and height near the coast. As the ocean depth increases,the tsunami wave width increases however the wave height decreases near the coast.

Fig. 7. Tsunami wave velocity and height at d = 20, t = 6 and various coast slopes.

Fig. 8. A comparison of tsunami wave height in deep water and near the coast at t = 4 and various ocean depths.

5.Conclusions

In this paper, a robust scheme based upon the method of lines has been introduced for simulating tsunami wave's propagation in the open ocean and on the ocean coast. The numerical results show that the proposed scheme display a high accuracy and reliability in solving such models. This numerical study was undertaken to examine the effect of coast slope and ocean depth on tsunami run-up. A nonlinear shallow-water model is employed for the numerical simulations. The general conclusion of this study is that the tsunami wave velocity diminishes and the height increases in the first moments and then decreases over time after entering the shoaling water of coastlines in its path. Consistent with previous studies and tsunami physics, it is found that the coast slope is of primary importance for determining the run-up height. The velocity and height of the tsunami wave decrease as the coast slope becomes steep. Another important conclusion is that the maximum wave height decreases near the coast as the ocean depth increases.

Acknowledgments

The author thanks the reviewers for their careful reading and helpful suggestions. The work is supported by PSRC(A Project Funded by the Basic Science Research Center of Majmaah University , KSA) and Project No. 60/38 .