# ANALYSIS OF STABILITY AND ACCURACY FOR FORWARD TIME CENTERED SPACE APPROXIMATION BY USING MODIFIED EQUATION.

Byline: Tahir.Ch, N.A.Shahid, M.F.Tabassum, A. Sana and S.Nazir

Abstract: In this paper we investigate the quantitative behavior of a wide range of numerical methods for solving linear partial differential equations [PDE's]. In order to study the properties of the numerical solutions, such as accuracy, consistency, and stability, we use the method of modified equation, which is an effective approach.To determine the necessary and sufficient conditions for computing the stability, we use a truncated version of modified equation which helps us in a better way to look into the nature of dispersive as well as dissipative errors. The heat equation with Drichlet Boundary Conditions can serve as a model for heat conduction, soil consolidation, ground water flow etc.Accuracy and Stability of Forward Time Centered Space (FTCS) scheme is checked by using Modified Differential Equation [MDE].

Key words: Accuracy, Stability, Modified Equation, Dispersive error, Forward Time Center Space Scheme.

1. INTRODUCTION

To analyze the simple linear partial differential equation with the help of modified equation, we consider one dimensional heat equation (Transient diffusion equation) which is parabolic partial differential equation.This equation describes the temperature distribution in a bar as a function of time. For converting this simple PDE into a modified equation, we use finite difference approximations by using the initial value conditions. This is obtained by expanding each term of finite difference approximation into a Taylor series,excluding time derivative, time - space derivatives higher than first order. Terms occurring in this MDE represent a sort of truncation error. These permit the order of stability and accuracy. With this approach, a modified equation, which is an approximating differential equation that is a more accurate model of what is actually solved numerically by the use of given numerical schemes.

To explain this scheme we are taking a long thin bar of homogeneous material.The temperature in a long thin bar must be insulated perfectly to maintain the flow of heathorizontally.

2. MATERIAL AND METHODS

2.1 Modified Equation

Modified Equation  is used MDE to analyze the accuracy and stability S

2.1 Modified Equation

Modified Equation  is used MDE to analyze the accuracy and stability of the solution originally solved by PDE's. This is obtained by expanding each term of finite difference equation with the help of Taylor Series. The general technique of developing modified equation for PDE's is presented by Warming and Hyett . We know that the general Linear PDE  is represented as

Equations

where c" is a real constant.The modified equation is used to deal with the numerical solution's behavior.

2.2 Difference Approximation

We are using here forward difference approximations 

Equations

2.4 Analysis of FTCS Approximationusing Modified Equation The modified equation for the FTCS approximation forf_t=

Equations

whichpredominately exhibits dissipative error . The lowest-order even derivate on the right-hand side of the modified equation (5) is 2. Therefore for the modified equation (5), the stability condition is C (2 = 2l) greater than 0, which implies that a greater than 0. However this yields no useful information since this parameter is chosen to be positive, and it is a coefficient in the original equation. However, if we implement the more general stability condition given by

Equations

3. RESULTS AND DISCUSSION

The amplitude of the exact solution decreases by the factor during one time step, assuming no boundary condition influence. The amplification factor is plotted in figure 2 for two values of r and is compared with the exact amplification factor of the solution. The diamond signs are the graph of G for r = 1/6, the solid line is the graph of |Ge| for r = 1/6, the plus signs are the graph of G for r=1/2 and the dashed line is the graph of |Ge| for r = 1/2. In this figure 2, we observe that the FTCS is highly dissipative for large value of AY where . As expected, the amplification factor agrees closer

Equations

4. CONCLUSION

MDE's in specific problems are more convenient for discussing the solution behavior, including physical interpretation, i.e. accuracy, stability and consistency. Many ordinary and higher order boundary value problems have been analyzed with the help of modified equation. The appropriate solution converges rapidly to accurate solution. So we say that MDE's are more beneficial for future use.

REFERENCES

 Randall J.L. Finite difference Methods for differential equation", University of Washington,USA, (2005).

 Lax P.D. and B. Wendroff Systems of Conservation Laws"Communications on pure and Applied Mathematics,13(2): 17-23(1960)

 Burden R.L. and J.A. Faires, Numerical Analysis" PWS Publishing Company, Boston,USA(2011).

 Richtmyer R.D. and K.W. Mortaon Difference Methods for initial Value Problems",John Wiley and Sons, New York, (1967).

 Brian Bradie A friendly introduction to Numerical Analysis" Pearson PrenticeHall,USA (2006).

 Morton K.W. and D.F. Mayers Numerical Solution of Partial Differential Equations" Cambridge University Press, Cambridge, (1994).
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