DEVELOPMENT OF A DISCRETE SIMULATION MODEL FOR TSUNAMI TIDAL WAVE

DEVELOPMENT OF A DISCRETE SIMULATION MODEL FOR TSUNAMI TIDAL WAVE OBAYOMI ABRAHAM ADESOJI Department of Mathematical Sciences, Faculty of Science, Ekiti State University, P. M. B 5363, Ado-Ekiti, Nigeria Copyright © 2018 Obayomi Abraham Adesoji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we develop new discrete models for the numerical simulation of non-linear ordinary differential equation arising from the dynamics of a tidal wave. A new set of non-standard schemes are proposed using the technique of non-local approximation. Both one step and two step formats have been considered. The schemes were found to be suitable for the numerical simulation of the Tsunami equation as proposed.


Introduction
One of the most important questions in prognostic tsunami modeling is estimation of tsunami run-up heights at different points along the coastline. Methods for numerical simulation of tsunami wave propagation are well developed and are widely used by a great number of scientists. To date, only a few existing numerical models have met current standards, and these models remain the only choice for use for real-world forecasts To some earlier modelers, the Tsunamis are assumed to be linear, long gravity waves at single frequencies. Several models that have proven very useful has employed non-linear views. Some of these works include: [1], [2], [3], [4], [5] and [6]. Since numerical applications has proven to produce alternative discrete model that can be used to approximate non-linear equations. We can apply numerical experiments to simulate various scenario possible for any given dynamical system A lot of work has been done in the area of numerical modeling of ordinary differential equations. Many of these techniques have been found to be useful for developing discrete solution for different type of equations.
Various researchers have come up with useful models in which the numerical simulations have been applied in the management and study of tidal waves among them are: [7], [8], [9], [10] and [11].
In this work we will use a combination of non-local approximation of the derivatives and the grid point estimates to develop one step and two step Non-standard finite difference schemes for the Tsunami equation.
Non-local approximations and renormalization of the denominator functions have been found to be appropriate for the solution of differential equations. The works of [12] and [13], [14], [15] and [16] have used these techniques extensively to develop discrete models that correctly follow the dynamics of the original differential equation. In many cases the schemes have been found to possess certain desirable qualitative properties like monotonicity of solutions. linear stability and preservation of the properties of the fixed points. The model of non-linear ordinary differential equation proposed in the book of [17] will be used for the numerical experiment. The significant of this numerical model is in the combination of several techniques in one scheme to simulate the original equation. Some denominator functions developed by these authors have been used for the purpose of comparison. These denominator functions have been developed based on the rule 2 of non-standard modeling technique proposed by [12].

The Tsunami model (Gill and Cullen 2005)
The Tsunami Model is given by y(x) > 0 is the height of the wave expressed as a function of its position relative to a point offsore. This is one of the simplest model for a tidal wave considering the complex dynamics of the phenomena being model. The function y(x) has a lot of underlining assumption and as such in reality may possess more complex properties. The equilibrium points of this equation are y= 0 and y=2; and 100 OBAYOMI ABRAHAM ADESOJI if y 0 = 2 the analytic solution is y(x) = 2sech 2 x Otherwise y(x) = 2sech 2 (x − c) (2) We will develop new schemes that possess the same qualitative properties as that of this differential equation.

Derivation of the Method
We will apply rule2&3 (see [12]) and their extensions in [14] to each of the components of the equation as shown below

Rule 2 (Mickens 1994)
Denominator function for the discrete derivatives must be expressed in terms of more complicated function of the step-sizes than those conventionally used. This rule allows the introduction of complex analytic function of h that satisfy certain conditions in the denominator .
It must be stated here that the selection of an appropriate denominator is an 'art' (Mickens 1999).
Close examinations of differential equation, for which the exact schemes are known, shows that the denominator function generally are functions that are related to particular solutions or properties of general solution to the differential equation. This therefore places great importance on the necessity of the modeler to obtain as much analytic knowledge as possible about the differential equation. A lot of such denominator functions have been developed by this author and many others, such will be used directly here.

Rule 3 (Mickens 1994)
The non-linear terms must in general be modeled (approximated) non-locally on the computational grid or lattice in many different ways.
Application of the combination of these two rules will give us the following transformations The following non-local approximations Sample renormalisation functions to be employed are

One step Schemes Scheme A
Applying non-local approximation to grid points using the transformation equations (3)and (9) in (1) we have the following We can choose any to form schemes of the form Applying non-local approximation to derivative using the transformation equations (3) in (1) we have the following We can choose any to form schemes of the form The direct substitution of a normalized denominator to replace h in the standard Finite Difference Scheme +1 = + ℎ ( , ), will result in the simple scheme given below in This scheme will not involve the application of any other Nonstandard modeling rule except the replacement of the denominator.

Two Step Schemes
Method I

Scheme C
Applying non-local approximation to the original differential equation using the transformation equations (5) in (1) we obtain the following We can choose any to form schemes of the form Note that = ℎ is the standard denominator function in Finite difference method We will also compute values for the scheme Definition 1: An initial value problem of a first order ODE can be represented as follows: y ′ = f(t, y) , у(t 0 ) = y 0 where y 0 is the value of y at time t 0 It is common fact to write the functional dependence +1 on the quantities , and h in the form; Where φ( , ; ℎ) is called the increment function.
Let us denote the approximate solution of (31) at grid point : Theorem 2: (Anguelov and Lubuma(2003): Assume that the difference scheme (32) is stable with respect to monotone dependence on initial value. Assume also that for every ℎ˃0 the equations 104 OBAYOMI ABRAHAM ADESOJI y = F(h ; y) and f(y) = 0 (36) in y have the same roots considered their multiplicity. Then the difference scheme (33) is stable with respect to monotonicity of solutions.
In the next subsection section we shall use the above theorems (1 -3) to establish the stability or otherwise of the non-exact schemes developed for the Logistic and Combustion equations. Please note that a major advantage of having an exact scheme for a differential equation is that questions related to the usual considerations of consistency, stability and convergence do not arise (see Mickens 1994).
In this Section we show that our schemes satisfy the sufficient condition for the stability properties described above. However the Scheme given by

Stability of Scheme
Clearly satisfy the conditions of theorem 2 and we conclude that the scheme Direct has the property of monotonicity of solutions The above also confirm Elementary Stability as stated in Theorem 3.         Surprisingly the direct substitution of h with normalized denominator function perform better than any of the other schemes in the long run (see the errors of scheme (DIRECT) marked green in fig. 13). This Scheme also possess the three stability properties examined. This is a