OPTIMAL CUBIC SPLINE METHOD FOR CONVECTION DIFFUSION EQUATION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In the proposed work, we present a new collocation technique based on cubic splines to solve initialboundary value parabolic partial differential equation. To attain fourth order accuracy, the proposed method requires only three spatial grid points as compare to the requirement of five grid points in the literature, using the collocation methods based on splines. We have used two stage Gauss Legendre method in time direction. An analysis has been done to prove the unconditional stability of the technique. To show the better accuracy of our method, numerical experiments are done by taking some examples from the literature. The results obtained, show the efficiency and the order of accuracy of the technique.


INTRODUCTION
In this paper, we are concerned with finding a stable and high accurate method for the approximate solution of a second order linear parabolic partial differential equation: (1) where Pu(x,t) = κu xx (x,t) + εu x (x,t) + ηu(x,t) is a linear second order operator and κ, ε and η are constants and κ > 0.
Collocation method is one of the popular methods used for finding the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs). In the collocation method, the approximate solution for differential equation lies in an approximation space of piecewise polynomials, with respect to the partition of the domain of the problem. Orthogonal piecewise polynomial collocation and spline collocation methods are few of them. The choice of spline functions as the piecewise polynomials results in a solution that is smooth at the nodes of the partition up to a certain degree. In literature, there are two typical ways for introducing collocation methods based on splines for the discretizing parabolic differential equations in spatial direction. One of these is standard second order formulation which requires the solution of a tridiagonal system of equations at each time step. Second way is to use a fourth order accurate deferred-correction or extrapolated spline collocation methods, which requires the solution of either a pentadiagonal system or two tridiagonal systems of equations at each time step. Archer [1] proposed cubic spline collocation method for quasi linear parabolic equations, which gives rise to fourth order pentadiagonal matrix structure. Mittal and Jain [6] used a combination of cubic spline collocation (CSC) and Crank-Nicolson (CN) to solve a convection diffusion equation which resulted in second order accurate solution. Various methods based on splines have been discussed to solve (1) such as cubic trigonometric B-splines [9], combination of spline and finite element [10], cubic spline [11], quartic and quintic B-spline [12] and exponential B-splines [13]. High order finite difference method for non linear parabolic partial differential equations are discussed in [14]. Non-polynomial cubic spline methods for the solution of parabolic equations are presented in [15].
In [2,7], a combination of quadratic spline collocation (QSC) and finite difference method In this work, we propose a CSC-GL4 algorithm for linear parabolic PDEs which is based on CSC method and fourth order two stage Gauss Legendre method. This algorithm gives fourth order accuracy in both the space and time variables with minimal computational efforts.
Stability analysis is done in an analytical manner in comparison to Christara et. al. [2] and [4], which proved stability in numerical manner. Since the proposed CSC-GL4 method uses triadiagonal system to find the solution of parabolic equation with high order accuracy in both time and space direction. So, it is better than QSC method in terms of computational time and efforts.
The paper is organized in the following manner. In Section 2, fourth order the cubic spline collocation method is proposed which gives rise to a tridiagonal structure. In Section 3, two stage Gauss-Legendre Method (GL4 Method) of order four is presented. In Section 4, stability of the CSC-GL4 algorithm is shown. In Section 5, numerical experiments has been done to illustrate the effectiveness of the algorithm.

FOURTH ORDER CUBIC B-SPLINE COLLOCATION METHOD
Let us consider is a cubic polynomial for 1 ≤ j ≤ J.
In this paper, we use the collocation form of cubic B-spline to approximate the exact solution u(x,t) of the differential equation. Any cubic spline function U(x,t) defined on the partition Ω can be written as a linear combination where s j (t) are time dependent parameters and {Ψ 3,−1 , Ψ 3,0 , Ψ 3,1 , . . . , Ψ 3, j , Ψ 3,J+1 } are cubic polynomial that forms a basis for the space of all cubic spline over the partition Ω. The cubic spline basis functions Ψ 3, j (x) at the nodes are defined as (Boor [3]): The values of Ψ 3, j (x) and its derivatives at different grid points obtained from (5) are given in the table 1.
The approximate value of the solution U and its two derivatives D x U and D 2 x U at the node x j in terms of parameters s j ≡ s j (t) are as follows: for j = 0, 1, . . . , J.
Fourth order accurate approximate solution of the parabolic partial differential equation (1), is obtained by following cubic spline collocation method where with U = u on the boundary of the domain.
To obtain the approximate solution U(x,t) at a particular grid point x j , we need to evaluate the parameters s j (t). This can be done by using the given boundary conditions and the collocation form (9) of the differential equation. The method (9), using equations (10)-(11) can be written as an equation with six unknown time dependent parameters s j−1 , s j , s j+1 ,ṡ j−1 ,ṡ j andṡ j+1 whereṡ j represents the derivative of s j at the knot x j with respect to time. To get rid two extra parameters we make use of boundary conditions. Using equations (3) and (6), we get (12) These equations allows us to eliminate s −1 , s J+1 ,ṡ −1 andṡ J+1 from the system of equations (9). So, we get a system of J + 1 equations in J + 1 unknowns.
Next, we will show that the numerical solution obtained by the method (9) is of order four.
For this, let us consider a cubic spline interpolant V of u defined by Theorem 1. Let V be a cubic spline interpolation of the exact solution u ∈ C 6 [a, b] which satisfies (13)- (14). Let U be as defined by equations (9)- (11). Then for 0 ≤ j ≤ N, Proof. The proofs (i)-(iv) follows from [5].
From Theorem 1, it can be observed that the CSC method (9) for the partial differential equation (1) is fourth order accurate.

TWO STAGE GAUSS-LEGENDRE METHOD(GL4 METHOD)
The two-stage Gauss-Legendre method of order four (GL4) is a particular class of implicit  Let ∆t be the step size in time direction so that t n = n∆t for n = 0, 1, . . .. In this method, solution of the initial value problem where, the weights w 1 and w 2 are calculated by After the initial vector C 0 has been found from the initial conditions, the approximate solution of the differential equation (1) at any time t can be obtained using the recurrence relation (20).
The CSC method together with the GL4 method produces an error of order O(h 4 + ∆t 4 ).

STABILITY OF CSC-GL4 METHOD
In this section, we will discuss the stability of the CSC-GL4 method (Cubic spline collocation in space direction and two stage Gauss Legendre method in time direction). The cubic spline collocation method given by equations (9) for the differential equation (1), in matrix form can be written as where D 1 and D 2 are square matrices defined as follows Here, B 0 , B 1 and B 2 are tri-diagonal square matrices of order J + 1, given by F is a column vector of order J + 1 corresponding to the given forcing function f and the boundary values. C is a column vector given by In homogeneous case, the two-stage cubic spline Gauss-Legendre method can be written as Using equations(21), the weights w 1 and w 2 are obtained from the equations (26) The system of equations (26) can be written as where υ 1 = 0.5 0.5 − 1 √ 3 and υ 2 = 0.5 0.5 + 1 √ (27), we get

. From the equations (25) and
Here, I is an identity matrix of order J + 1. We can further write the equation (28) as Since the matrix D 2 is diagonally dominant, it is invertible. So, from the equation (29) we obtain (30) To prove that the CSC-GL4 method is stable, it is sufficient to prove that the eigen values of the matrix X are less than the eigen values of the matrix Y, where the matrices X and Y are given by When the mesh size increases, i.e. h → 0 the matrix D −1 2 D 1 → κB −1 0 B 2 . It can be easily seen that the eigen values of the matrix B −1 0 B 2 are negative. Let λ be an eigen value of the matrix D −1 2 D 1 . Then the eigen values of the matrix X corresponding to the same eigen vector are given by the characteristic equation and the eigen values of the matrix Y are given by the equation It can be easily seen that Max(eig(X)) < Min(eig(Y )). Thus the proposed CSC-GL4 method is unconditionally stable.

NUMERICAL EXAMPLES
In this section, we implement the proposed CSC-GL4 method to compute the approximate solution of some test problems. To show the accuracy of the method, we have calculated the maximum absolute error between the exact and approximate solution by the formula, L ∞ = max j |u j −U j |. Order of convergence of the proposed method is calculated using the following formula: where L ∞ (J 1 ) and L ∞ (J 2 ) are the errors for number of grid points J 1 and J 2 respectively. Example 1. Consider the test problem from Liu and Wang [4], A comparisons of the errors with the results obtained in [4] are also done in the tables 3 and 4.
In table 5, errors are calculated for κ = 0.5 at time T = 5 . Fourth order accuracy can be seen for h = k.
Example 2: Let us consider the differential equation considered in [2] subject to the initial condition The exact solution of the problem is given by We Mittal and Jain [6] and Ismail at el. [8]. From the comparisons done in the table, we can see that our method works better compared to Mittal and Jain [6] in every possible way but sometimes it is less accurate than Ismail at el. [8]. However, our method is unconditionally stable compared to conditionally stable method of Ismail at el. [8]. To show the fourth order accuracy of the method we have calculated the errors for various and listed in table 9. Graphical representation of numerical solution obtained is shown in figure 3.
Errors have been computed at various time levels for h = k = 0.01 and comparisons with the results listed in Mittal and Jain [6] and Ismail at el. [8] have been done in table 10. We can see that our method works better.

CONCLUSION
In this work, we proposed a new CSC-GL4 method to solve initial-boundary value parabolic partial differential equation. A compact collocation method based on cubic splines is used in space direction. Discretization in time direction is handled by using two stage Gauss Legendre method. To attain high order accuracy the proposed method requires only three spatial grid points at each time step as compare to the requirement of five grid points at each time step in the literature, using the collocation methods based on splines. The proposed method is of order