On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations

: This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature


Introduction
The time-dependent diffusion equation of the form (1) describing a damp diffusion in time is considered and it is called a second order parabolic partial differential equation.If the equation (1) above is given the following initial and boundary conditions (2) and respectively, the equation ( 1)-( 3) becomes an initialboundary value problem and the solution can be obtained using numerical methods.Numerical methods is a method of obtaining an approximate solution to partial differential equations whose solution cannot be obtain analytically.There are different types of numerical methods but for the purpose of this work, we shall consider the explicit (FTCS); Implicit (BTCS) and Du Fort-Frankel schemes.The efficiency of the numerical schemes depends solely on their stability.Lot of researchers have worked on finite difference methods for parabolic partial differential equations, among them are; Crank J and Philis N. [2] worked on practical method for evaluating numerical of solution of partial differential equation of heat conduction type.Recktenwald G. W [4] discussed the three finite difference methods (FTCS; BTCS and Crank-Nicolson methods) to solve one dimensional boundary problem.Karatay I. and Bayramoglu.S [5] obtained the solution of time fractional heat equation using Crank-Nicolson method.Aswin V. S et al [6] described three different numerical schemes to approximate the solution of the convection-diffusion equation.Azad T. M. A. K and Andallah I. S. [7] studied stability analysis for two standard finite difference schemes, forward time and centered space and centered space (FTBSCS) and forward time and centered Parabolic Partial Differential Equations space (FTCS) for convection-diffusion equation.Olusegun O.A et al [9] solved the one dimensional heat equation using the explicit scheme.Adak M. and Mandal N. R [13] solved the transient heat equation with convection boundary condition using explicit finite difference scheme.Adak.M [14] studied the effect of explicit and implicit schemes on one dimensional diffusion equation with dirichlet boundary condition.There are also some interesting texts for the subject, they are Williams F. Ames [10], Smith G. D [11] and Grewal B. S [12].
The main objectives of this paper is to compare the stability of the schemes using von Newmann method.Also, their results when applied to a parabolic equations are also compared.

Problem Definition and Methodology
The one dimensional heat equation ( 1) -(3) of length rod is considered, where temperature, = time, = length and is the thermal diffusivity, also = thermal conductivity, c = heat capacity and density.For the derivation of the scheme, we shall use the following derivative in [3,11], All derivatives in the equation (1) are approximated using Taylor's series expansion.

The Explicit Scheme
The explicit scheme is derived from equation (1) by replacing the first order derivative and second order derivative by forward difference in time and central difference in space respectively.The derivation is shown below: , then equation ( 5) becomes

Local Truncation Error
The local Truncation error of the explicit scheme has it principal part as Therefore, its local truncation error is ( ) + (ℎ ).

Stability Condition (Using von Newman Method)
von Newman stability method is the most widely used procedure for determining the stability of finite difference approximation (Lapidus Leon).The method introduces an initial line of errors as represented by Fourier series and consider the growth of these error as increases.The stability of the explicit method using von Newmann method is shown as follows; the explicit scheme is given by the equation above in error form is written as let / $,% = 0 123 0 $45 = 6 7 0 $45 where 6 7 = 0 123 (8) substituting equation ( 8) into equation ( 7) gives using simple mathematical principle and on cancelation of common terms, equation ( 9) yields which can be written as using the following trigonometry identities using equation ( 11) in ( 12) we get The necessary and sufficient condition for the error to bounded, keeping to numerical stability is |6| ≤ 1 therefore, the condition for stability of the explicit scheme will be considering equation (13) we have that shows that the explicit scheme is conditionally stable.

Dufort-Frankel Scheme
The derivation of the Dufort-Frankel approximation is simply by replacing the first and second order derivatives in equation ( 1 which can be wriiten as Which further gives where * = ' ( #) equation ( 20) can be written as Equation ( 21) is called the Dufort-Frankel finite difference approximation.It can also be written more explicitly as

Local Truncation Error
The

Stability Condition (Using von Newmann)
The Dufort-Frankel scheme is given by equation ( 21), rewriting equation (21) in error form gives; ( using the same procedure as in the stability criterion for the explicit scheme, we have in equation ( 24), we have substituted equation (8) solving equation ( 27) and using the fact that 89: ; = 1 − :<= ; , we get Which gives

Implicit Scheme (BTCS)
The implicit scheme is derived by replacing the first order derivative by forward difference and the second order derivative by central difference with the I + 1.The procedure is as follows: we have Equation ( 29) is the implicit approximation.The equation leas to a tridiagonal system.Parabolic Partial Differential Equations

Local Truncation Error
The local truncation error of the implicit scheme is given by , ℎ ).

Stability Condition (Using von Newmann)
The stability of theimplicit scheme using von Newmann method is given by for all * > 0, and all ; .It is observed that 0 < 6 ≤ 1. Showing that the scheme is unconditionally stable.

Numerical Examples
This section presents some numerical examples on the comparison of the finite difference schemes.
Consider the following mathematical model subject to the initial condition ( , 100) = 100 (32) and boundary conditions (0, ) = 0 = (1, ), > 0 (33) of the temperature distribution in a rod of length = 1W with its end point at 0 X .and initial point at 100 X Y.Given that the analytical solution of the model is

Solution:
Solving first with the explicit scheme, we use equation ( 6) to obtain the values for 1 ≤ < ≤ 9 and the various steps from 0 ≤ I ≤ 9 and the results are presented in the following table, see [1] for few steps on the solvings.Solve the following heat equation subject to the initial condition ( , ) = :<=^ , [0,1G

Discussion
Table 1 presents the results of both explicit and Dufort-Frankel scheme, from our calculation at * = & in the explicit scheme, it results into Bender-Schmidt scheme, also, the Dufort-Frankel scheme also work like the explicit scheme at * = & , hence the same result is presented for the two scheme at * = & .Table 2 shows the comparison of the numerical solutions of the explicit schemes and the exact solutions, the percentage errors are also presented.Tables 3 and 4, shows the results of the implicit scheme and the comparison of its numerical solutions with the exact solutions.From tables 2 and 4, it can be observed that the implicit scheme is good compare to the other two schemes.Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes.

Conclusion
From the results, it is observed clearly that the implicit scheme is efficient and fast in implementation than the other two schemes.Also, the implicit scheme and Dufort-Frankel schemes as seen using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable.This can be seen in example 2, where both schemes perform better than the explicit scheme and also the implicit scheme performs better than the Dufort-Frankel scheme.The results of the methods agree with existing findings in literature, see Omowo B. J and Abhulimen C. E [3], Olusegun O. A, Hoe Y. S, Ogunbode E. B [9] and Adak M. [14] that the implicit scheme has no restriction for the value of its mesh ratio and that smaller time steps produces more accurate results.

If = 0. 1
and * = & , then the problem using explicit, Dufort-Frankel and Implicit schemes and compare the numerical solutions with the exact solutions at = 0.4.

Table 2 .
Comparison of the explicit scheme with the exact solutions.

Table 3 .
Results of example 1 using Implicit scheme.

Table 4 .
Comparison of the Implicit scheme with the exact solutions.