Stability Control of Stretch-Twist-Fold Flow by Using Numerical Methods Shahab

In this study, the multistep method is applied to the STF system. This method has been tested on the STF system, which is a three-dimensional system of ODE with quadratic nonlinearities. A computer based Matlab program has been developed in order to solve the STF system. Stable and unstable position of the system has been analyzed graphically and finally a comparison as well as accuracy between two-step sizes with detail. Newton’s method has been applied to show the best convergence of this system.


Introduction
The distinction between slow and fast dynamos was first drawn by Vainshtein & Zeldovich (1972) in this research; we describe the stretch-twist-fold (STF) fast dynamo, which is the archetype of the elementary models of the process.Basically, stretch-twist-fold is applied in fluid mechanics in aerospace.In space, any fluid can be D-Tracked easily so a magnetic field is required to compel the fluid to be in the same orbit and this method is called STF system.In this paper, we will investigate the accuracy of numerical method.The Multistep method was first introduced by Goldstine, Herman H. in the beginning of 1977's.This iterative method has proven rather successful in dealing with various scientific problems [1][2][3][4] since it provides analytical solutions, which is a standard numerical method.This method has also been applied to solve nonlinear systems of ordinary differential equations.For example, H. B. Keller [5] presented an extensive comparative study on the accuracy of the multistep method and C. Lubich [6] studied the effects of time steps on the stiff problem.J. O. Fatokun and I. K. O. Ajibola [7] studied multistep method for integrating ordinary differential equations on manifolds.Differential equations are used to model problems in science and engineering that involve the change of some variables with respect to another.Most of their problems require the solution to an initial-value problem that is the solution to a differential equation that satisfies a given initial condition.In most real-life situations the differential equation that models the problem is too complicated to solve exactly and one of two approaches is taken to approximate the solution.The first approach is to simplify the differential equation to one that can be solved exactly and then use the solution of the simplified equation to approximate the solution to the original equation.The other approach, which we will examine in this paper, uses methods for approximating the solution of the original problem.This is the approach that is most commonly taken, since the approximation methods give more accurate results and realistic error information.The objective of this research is to solve STF system and test nonlinear behavior with different time steps.This modified method is able to find a stable and unstable position of STF system.This method can also give the exact values after iteration results.Newton's method is able to show the best convergence than fixed point iteration method.

Stretch-Twist-Fold Flow (STF)
The STF flow is defined as where α = 0.1, β = 1 are positive real parameters and related to the ratios of intensities of the stretch, twist and fold ingredients of the flow.

Description of Methods
The methods we consider in this section do not produce a continuous approximation to the solution of the initial-value problem.Rather, approximations are found at certain specified and often equally spaced points.Some method of interpolation is used if intermediate values are needed.We need some definitions and results from the theory of ordinary differential equations before considering methods for approximating the solutions to initial-value problems.
Definition 3.1: A function   , f x y is said to satisfy a Lipschtiz condition in the variable y on a set This first part of this section is concerned with approximation the solution   y x to a problem of the form has a unique solution   y x for a x b   .The methods of Euler and Runge-kutta are called onestep methods because the approximation for the mesh point 1 i x  involves information from only one of the previous mesh points i x although these methods can use functional evaluation information at points between i x and 1 i x  , they do not retain that information for direct use in future approximations.All the information used by these methods is obtained within the subinterval over which the solution is being approximated.Since the approximate solution is available at each of the mesh points 0 1 , , , i x x x  before the approximation at 1 i x  is obtained and because the error    tends to increase with I, it seems reasonable to develop methods that these more accurate previous data when approximation the solution at 1 i x  .Methods using the approximation at more than one previous mesh point to determine the approximation at the next point are called multistep methods.Definition 3.2: An m-step multistep method for solving the initial-value problem (3.1) is one whose difference equation for finding the approximation When b -1 = 0 then the method is called explicit or open.Since Equation (A) then gives x y x y x y  Equation (B) becomes   ,  , .

Unstable Position
When α = 1, β = 0.1 and h = 0.01 then we can determine the unstable position of the system that is shown in Table 1 and easily analyzed by the Figure 1.

Stable Position
When α = 0.1, β = 1 and h = 0.001 then we can determine the stable position of the system that is shown in Table 2 and easily analyzed by the Figure 2.

Fixed Points for Function of Several Variables
In this section, we will discuss about fixed point iteration method and Newton's method.
A system of nonlinear equations has the form , , , Here each function f i can be thought of as mapping a vector The system of n nonlinear equations in n unknowns can alternatively be represented by defining a function f, mapping , , , .
In an iterative process for solving an equation   0 f x  was developed by transforming the equation into one of the form   x g x  .The function g is defined to have fixed points precisely at solutions to the original equation.A similar procedure will be investigated for function from  Consider the STF system (see Figure 3):  Now we have tested this system in Newton's method and comparison with fixed point iteration results.
Newton's method for systems, like the one-dimensional Newton's method, a fixed point iteration based on a linearization of  .
then the Taylor series for   f x has the form Newton's method is derived just as it was for the one-dimensional case: neglecting the remainder term, we have This is Newton's method for systems.
Where,   In practice it is preferable to solve and then add this quantity to k x we have By system (C), the Jacobi matrix   J x for this system is The results are given in Table 3.
According to previous examples, we can easily analyze that Newton's method is more accurate than fixed point iteration method.

Conclusion
In this paper, MATLAB programming has been used to solve the STF system with variable time steps (∆t = 0.01, 0.001).We have obtained good results by using two methods applied to the STF system concerning the system is stable and unstable state.The modified method was computed by developing simple algorithm without perturbation techniques i.e. linearization or discretization.In all the considered cases, it has been proved that the modified multistep method appears to be the best method to approximate this solution based on its accuracy and Newton's method is a good example to solve root finding problem in STF system.Newton's method is able to show the best convergence than fixed point iteration method.

Acknowledgements
The first author is very thankful to all of his co-authors and especially to Professor Shu Yonglu for advising and giving me the opportunity to conduct this research and also very much thankful to Sustainable Energy Technologies Centre, King Saud University for funding the research.

1 ,
then the method is called implicit or closed.Since 1 i y  occurs on both sides of Equation (A) and is specified only implicitly.To begin the derivation of the multistep methods, note that the solution to the initial-value problem (3.1), if integrated over the interval   x without knowing   y x the solution to the problem, we instead integrate an interpolating   L x to     , f x y x that is determined by some of the previously obtained data points

Figure 3 .
Figure 3. Portrait and x, y, z direction of STF system.