Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.


Introduction
In this paper, we present an implicit non-polynomial spline functions based scheme for the numerical solution of the following one dimensional convection-diffusion equation, ( , ), 0 1, 0 subject to the boundary conditions 0 1 (0, ) ( ), (1, ) ( ), 0 u t g t u t g t t t (2) and with the initial condition ( ,0) ( ), 0 1 u x f x x d d (3) where 0 H ! , is the phase speed and 0 J , is the viscosity coefficient.In natural systems, diffusion is one of the important mechanism which means that the mean square displacement of a diffusion particle is in a linear relation with time.The convection-diffusion equation describes the various physical phenomena where energy is transformed inside a physical system due to the combination of convection as well as diffusion processes.
It arises in various fields of applied sciences and engineering such as oil reservoir simulations, transport of mass and energy, global weather production, dispersion of diffusion process, due to these vide variety of physical implementations a great deal of researches have been done for the numerical and closed form solutions of convection diffusion type equation for eg., Jain and Aziz [4] used the adaptive spline function approximation for the numerical solution of convection-diffusion equation.
Mohammadi [5] applied exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet's type boundary conditions.Caglar et.al. [6] presented B-spline solutions for a convection-diffusion equation.Ravi Kanth and Aruna [1] implemented the differential transform method for solving linear and nonlinear Klein-Gordon equation.Mohammadi [2] presented the exponential spline approach and Lin [3] constructed the parametric cubic spline method for the numerical solution of non-linear Schrodinger equation.

Derivation of the Numerical Scheme
For the positive integers N and M , let the partition of , x Z . Each segment can be written as [7].

P x t P x t
The equations ( 4) and ( 6) yield the relation where As w 0 o , Eqn. ( 7) transformed into ordinary cubic spline relation as in [7].Considering the equation ( 7) at th j and ( 1) th j time levels, so that we can get, , 1, 2,3, , 1

Remark :
The truncation error for Eq. ( 8) is x t equation ( 1) can be rewritten in discretized form as 1 To obtain the approximation for Eq. ( 9), based on Taylor series using the following finite difference approximations

Stability analysis
The stability has been proven through the Von Neumann technique,the numerical solution can be expressed by means of a Fourier series exp where h V I .After some manipulations in Eqn. (15)  and then using the Euler's formula, we get For stability as the time increases, we want the amplification factor j [ must satisfy 1 and for this we must have,

Numerical results
In this section, in order to demonstrate the accuracy and effectiveness of the proposed scheme a few numerical evidences are given, The L f and 2 L errors along with the order of convergence are reported in Table 1, which shows that the purposed method is having fourth order convergency

I
is the wave number and j [ is the amplitude at time level j .Substituting Eqn.(14) in Eqn.

Figure 2
and 4 shows the absolute errors for Examples 1 and Example 2 respectively, and it can be observe from these figures that our method supports the theoretical results.

Table 2
, represents the absolute errors for Example 2 at different spatial and time levels.From this table we can observe that the numerical results are in excellent agreement with the analytical solutions.Figure1and 3 exhibits the numerical solutions for Examples 1 and Example 2, respectively.It is noticed that our numerical results are in good accordance with the analytic solutions.

Table 2 .
Absolute errors for Example 2