Exponential B-Spline Solution of Convection-Diffusion Equations

We present an exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet’s type boundary conditions. The method is based on the Crank-Nicolson formulation for time integration and exponential B-spline functions for space integration. Using the Von Neumann method, the proposed method is shown to be unconditionally stable. Numerical experiments have been conducted to demonstrate the accuracy of the current algorithm with relatively minimal computational effort. The results showed that use of the present approach in the simulation is very applicable for the solution of convection-diffusion equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithm is seen to be very good alternatives to existing approaches for such physical applications.


Introduction
Convection-diffusion equation plays an important role in the modeling of several physical phenomena where energy is transformed inside a physical system due to two processes: convection and diffusion.The term convection means the movement of molecules within fluids, whereas, diffusion describes the spread of particles through random motion from regions of higher concentration to regions of lower concentration.Also this equation describes advection-diffusion of quantities such as heat, energy, mass, etc.They find their applications in water transfer in soils, heat transfer in draining film, spread of pollutants in rivers, dispersion of tracers in porous media.They are also widely used in studying the spread of solute in a liquid flowing through a tube, long range transport of pollutants in the atmosphere, flow in porous media and many others [1][2][3][4].
We consider the initial-value problem for the one-dimensional time-dependent convection-diffusion equation 2 2 ,0 ,0 , subjected to the initial conditions and with appropriate Dirichlet boundary conditions where the parameter  and is the viscosity coeffic is the phase speed both are assumed to be po ient and sitive.
     0 , x g t  and   1 g t are known functions with suffient smoothness.
ssary to calculate the transport of fluid properties or trace con ci It is nece stituent concentrations within a fluid for applications such as water quality modeling, air pollution, meteorology, oceanography and other physical sciences.When velocity field is complex, changing in time and transport process cannot be analytically calculated, and then numerical approximations to the convection equation are indispensable [4].They are also important in many branches of engineering and applied science.Therefore many researchers have spent a great deal of effort to compute the solution of these equations using various numerical methods.There are many studies on the numerical solution of initial and initial-boundary problems of convection-diffusion equation [1, include finite difference methods, Galerkin methods, spectral methods, wavelet-based finite elements, B-spline methods and several others.These equations are characterized by no dissipative advective transport component and a dissipative diffusive component.When diffusion is the dominant factor, aall numerical profiles go well.On the contrary, most numerical results exhibit some combination of spurious oscillations and excessive numerical diffusion, when advection is dominant transport process.These behaviors can be easily explained using a general Fourier analysis, little progress has been made to overcome such difficulties effectively.Using extremely fine mesh is one such alternative but is not prudent to apply it as it is computationally costlier.So a great effort has been made on developing the efficient and stable numerical techniques.
Nguyen and Reynen [5] presented a space-time leastsquares finite-element scheme for advection-diffusion eq uests as go m asis for the space of exponential spline is given an uation.Codina [7] considered several finite-element methods for solving the diffusion-convection-reaction equation and showed that Taylor-Galerkin method has a stabilization effect similar to a sub grid scale model, which is in turn related to the introduction of bubble functions.Dehghan [12] developed several different numerical techniques for solving the three-dimensional advection-diffusion equation with constant coefficients and compared them with other methods in literature.These techniques are based on the two-level fully explicit and fully implicit finite difference approximations.Dehghan [13] developed a new practical scheme designing approach whose application is based on the modified equivalent partial differential equation (MEPDE).Dehghan and Mohebbi [20] presented new classes of high-order accurate methods for solving the two-dimensional unsteady convection-diffusion equation based on the method of lines approach.These methods are second-order accurate and techniques that are third order or fourth order accurate.Dehghan [14] derived a variety of explicit and implicit algorithms based on the weighted finite difference approximations dealing with the solution of the one-dimensional advection equation.Dehghan and Shakeri [21] obtained the solution of Cauchy reaction-diffusion problem via variational iteration method.
It is well-known that a good interpolating or approximating scheme, in addition to the standard req od approximation rate, low computational cost, numerical reliability, should possess the capability of reproducing the shape of the data.It has been found in the literature that using piecewise polynomial functions leads to better convergence results and simpler proofs than using polynomials.Recently, Spline and B-spline functions together with some numerical techniques have been used in getting the numerical solution of the differential equations.Kadalbajoo and Arora [4] used Taylor-Galerkin methods together with the type of splines known as Bsplines to construct the approximation functions over the finite elements for the solution of time-dependent advec-tion-diffusion problems.Mittal and Jain [1] discussed collocation method based on redefined cubic B-splines basis functions for solving convection-diffusion equation with Dirichlet's type boundary conditions.The main objective of this study is to develop a user friendly, economical and stable method which can work for higher values of Péclet number for convection-diffusion equation by using redefined cubic B-splines collocation method.
In the current paper, we develop the collocation method by using the exponential B-spline function for nu erical solution of the convection-diffusion equation.Our main aim is to improve the accuracy of B-spline method by involving some parameters, which enable us to obtain the classes of methods.Our method is a modification of cubic B-spline method for solution of (1).Application of our method is simple and in comparison with the existing well-known methods is accurate.While solving initial boundary value problems in partial differential equations, the procedure is to combine a spline approximation for the space derivative with a Crank-Nicolson finite difference approximation for the time derivative.The combination of a finite difference and an exponential spline function techniques provide better accuracy than the finite difference methods.Therefore, the time derivative is replaced by finite difference representation and the first-order space and second-order space derivatives by exponential B-spline.Also, we study stability for the new method and will show that it is unconditionally stable.We use the exponential B-splines basis that leads to the tri-diagonal system which can be solved by the well known algorithm.Numerical examples are presented which demonstrate that the present scheme with exponential B-splines gives more accurate approximations than the scheme using cubic B-splines.Similar to the method in [1] our method is a user friendly, economical and stable method which can work for higher values of Péclet number for convection-diffusion equation.
This paper is arranged as follows.In Section 2, the Bsplines b d some interpolation results for the exponential spline interpolate are stated.In Section 3, the construction of the exponential B-spline collocation method for the solution of the convection-diffusion equation is described.The stability analysis of the presented method is discussed in Section 4. In Section 5, numerical experiments are conducted to demonstrate the efficiency of the proposed method and confirm it's theoretical behavior.These computational results show that our proposed algorithm is effective and accurate in comparison with the literature.Finally results of experiments and the conclusions are included in Section 7.
gives the cubic splines; while    implies that , whic  h gives the linear splines.
Following [25,26] the solution of the above boundary problem is value where s are prescribed non-negative real numb ong the various classes of splines, the polynomial Am spline has bee has mits of Bsplines w ned as follows.Let n received the greatest attention primarily because it admits a basis of B-splines which can be accurately and efficiently computed.It been shown that the exponential spline also ad a basis hich can be defi x and having a finite support on the four consecutive intervals According to McCartin [27], the , , additional knots outside the region, positioned at: and .

Each basis function is twice continuously differentiable. The values o and
The set of exponential B-splines where i  are time dependent parameters to be determined from the exponential B-spline collocation form of Equation (1).
ar to t anoth Simi he above theorem we have er companion result due to McCartin [29].
Theorem 2.2 These theorems together with the de Boor-Hall error estimate Prenter leads to the following corollary.

Numerical Method
The region   To apply the proposed method, discretizing the time derivative in the usual finite difference way and applying  .be the exponential B-splines Crank-Nicolson scheme to (1), we get currence relation, once the initial vectors have been com ted from the initial and boundary c on

Treatment of Boundary Conditions
In order to eliminate the parameters here is the time step.n for and putting the values of the nodal values derivatives using Equations ( 9)-( 11) at the knots i quatio (12) yields the following difference equation t ables ,

cosh sinh
The system (13) consists of linear equations in of unknowns To obtain a unique two additional constraints are required.These are obtained from the boundary conditions.Imposition of the boundary conditions enables us to eliminate the parameters .
solution to t is system Expa ng u in terms of approximate exponential ndi B-spline form from (9) at ula 0 Solving the obtained equations we get the values of

The Initial State
The initial vector n u x g u x g  

Stability Analysis
We have investigated stability of the presented method by applying von-Neumann method.Now, we cosider the trial solution (on Fourier mode out of the full solution) at a given point i x   exp i , where β the mode number, h is the element size and i 1   .Now, by substituting (16) in (13), and symlifying the equation, we obtain where , where P e is called Péclet number in (18), we get Since numerator in (19) is less than denominator, therefore

It means t . on h
, hence the method is unconditionally stable.
hat there is no restriction on the grid size, i.e and , but we should choose them in such a way that the accuracy of the scheme is not de-

Numerical Illustrations
In this section, some numerical examples are presented to evaluate the performance of the proposed method.We consider six convection-diffusion equations which the exact solutions are known to us.To illustrate our presented method and to demonstrate it's applicability computationally, computed solutions for different values of k graded.

, , k  and h
 are compared with ints and with the results in exis exact solutions at grid po ting methods.All prore run in Mathematica 6.0.The computed abaxsimume absolute errors in numerical solutions are listed in Tables.
For the sake of comparison, following [1], some important non-dimensional parameters in numerical analysis are defined as follows: Courant number: The Courant number is defined as grams a solute errors and m

Grid Péclet Number:
The Péclet number is defined as Numerical results confirm that the Péclet number is high, the convection term dominates and when the Péclet number is low the diffusion term dominates.
Example 5.1 Consider the c vection-diffusion equation [1,10] and subject to the initial condi The theoritical solution of this problem is The boundary conditions are obtained from the theoritical solution.
In all computations, we take

n-diffusion e
The boundary conditions are obtained from the analytical solution.
In all computations, we take and subject to the initial conditions The analytical solution of this problem is The theoritical solution of this problem bsolu r method an The a te errors of ou d the method in [1] for example 5.1.
Our method Mittal and   Mittal and Jain [1] and t or example 5.2.
Our method The boundary conditions are obtained from e theori-th tical solution.
In all computations, we take 0. lts indi s sim ar to tively, but the less than [3].cate that the errors in our [1] and [3] The analytical solutio of this pr n oblem is The boundary conditions are obtained from the analytical solution.
In this example, first similar to [24] we take L = 1, e P  The in sults s is the absolute errors for diffe are tab ble 7 and co pared with res lts in [1].The indicate th rrors in hod more ilar to [1 n we c ur meth ethods in [1,4] 10 bles sho eth ble side thi s a compari the method in [1].Dehghan [3] .The Ta w that our m od is sta be s our method i ccurate in son with te errors of i r example 5.3.    .The results are computed for different time levels and listed in Table 13.Also we comp our results e that the errors in our method more or less is similar to [1] but the errors in our method are much less than the errors in [9].

Th l solution o is
Example 5.6 Consider the following equation [1,23] are results with the results in [1,9].The indicat and subj tial cond ion ected to the following iniit

Discussions
an ex o solve the convection-diffusion equation with Dirichlet's type boundary conditions and has been efficiently illustrated.To tackle this, the prop ed scheme of exponential B-spline in space and the Crank-Nicolson scheme in time have been combined.By taking different va e tabulated in is with ethod is [1,23]


is an exponential spline of order four, lies in the is determined by the time evolution of the vector j n  which is found repeatedly by solving the re-

Table 1
(8)sing the expression(8)and Table1, nodal value and its first and second derivatives at the knots

Table 8
in

Table 11 . The absolute errors of our method for example 5.4.
e P 