Spline in Compression Methods for Singularly Perturbed 1 D Parabolic Equations with Singular Coefficients

In this article, we discuss three difference schemes; for the numerical solution of singularity perturbed 1-D parabolic equations with singular coefficients using spline in compression. The proposed methods are of accurate and applicable to problems in both cases singular and non-singular. Stability theory of a proposed method has been discussed and numerical examples have been given in support of the theoretical results.  2 2 O k h   1


Introduction
We assume that the functions, and are sufficiently smooth and their required high-order derivatives exist in the solution space . This class of problems arise in various fields of science and engineering, for instance, fluid mechanics, quantum mechanics, optical control, chemical-reactor theory, aerodynamics, geophysics etc.There are a wide variety of asymptotic expansion methods available for solving the problems of the above type.But there can be difficulties in applying these asymptotic expansions in the inner and outer regions, which are not routine exercises but require skill, insight and experimentations.In many applications Equation (1) represents boundary or interior layers and has been studied by many authors.Henrici [1] has described the discrete variable methods for ordinary differential equations.Ahlberg et al. [2] and Greville [3] have worked on the theory of splines functions and their applications.An introduction to singular perturbations was given by Malley [4].Abrahamsson et al. [5] have discussed the finite difference approximations for the system of singularly perturbed ordinary differential equations.Further Prenter [6], Boor [7], and Hemker and Miller [8] have studied various splines and variational methods to solve differential equations.A uniformly accurate difference method for a singular perturbation problem has been analyzed by Berger et al. [9].Further, Kreiss and Kreiss [10] and Segal [11] have discussed stable numerical methods for singular perturbation problems.Later, Jain and Aziz [12] have derived an efficient numerical method for the solution of convection-diffusion equation using adaptive spline function approximation.Miller et al. [13] have used piecewise uniform meshes for upwind and central difference operators for solving singularly perturbed problems.Kadalbajoo and Patidar [14] have studied the spline in compression methods for the solution of a class of singularly perturbed two point boundary value problems.Later, Mohanty et al. [15] have extended the work discussed in [14] and solved singularly perturbed two point singular boundary value problems.In 2005, Khan and Aziz [16] have discussed the tension spline method for the solution of second order singularly perturbed boundary value problems.Khan et al. [17] have made a survey on various parametric spline function approximations.However, the methods discussed in [17] are only applicable to problems in rectangular coordinates.In the past difficulties were experienced for the numerical solution of singularly perturbed one space dimensional parabolic problems in polar coordinates.The solution usually deteriorates in the vicinity of singularity.In recent past, Mohanty et al. [18] have derived new stable spline in tension methods for singularly perturbed one space dimensional parabolic equations with singular coefficients.In this paper, we have presented a new approach based on spline in compression to solve singularly perturbed parabolic equations of type (1).We have refined our procedure in such a way that the solution retains its order and accuracy even in the vicinity of the singularity x = 0.It is well known that the most classical methods fail when  is small relative to the mesh length h > 0, that is used for discretization of the differential Equation (1) in the x-direction.Our aim is to show that compression splines can furnish accurate numerical approximations of Equation ( 1), when all or any of the coefficients and  , , f x t contain singularity at x = 0 and when  is either small or large as compared with h.We consider three types of problems.In the first case, we analyze the problems in which the second derivative term xx u x and the function term x u x but lacking the function term are considered in the second case.Finally, the third case deals with the most general problems.In all cases, we use the continuity of first derivative of the spline function.The resulting spline difference methods are two-level implicit schemes (see Figure 1) and of accurate and are tridiagonal system of equations at each advanced time level, which can be solved by using a tri-diagonal solver.The main significance of our work is that the proposed compression spline difference schemes are applicable to both singular and non-singular problems.In Section 2, we  have discussed the derivation of the spline methods and their application to singular problems.In Section 3, we have discussed stability analysis of a method.In Section 4, numerical results of three different singular problems have been given to demonstrate the utility of the proposed method.The numerical results confirmed that the proposed compression spline methods produce an oscillation-free solution for 0 1    everywhere in the solution region 0 < x < 1, t > 0.

Description of the Compression Spline Method
The solution domain     We consider the following three cases: Case 1: First we consider the differential equation which is a particular case of Equation (1) in which the first derivative term   , u x t x For the derivation of the method for the Equation (4), we follow the approaches given by Kadalbajoo and Patidar [14], and Mohanty et al. [15]. is absent.

Now we consider the ordinary differential equation
jth -level Copyright © 2012 SciRes.

OJDM
The numerical solution of this equation is sought in the fo Now using the approximations (12a)-(12c) in Equation (11) and neglecting high order terms we pression spline scheme for the Equation fo rm of the spline function   S x , which on each inter- The interpolating conditions: and the continuity condition: Solving the Equation (6) and using co the interpolating nditions (7), we get Equation ( 9) is known as spline in compression, Replacing l by l + 1 in Equation ( 9), we can obtain the spline function Differentiating Equation ( 9) with respect to "x" and using the continuity condition (8), we obtain the spline in compression scheme for the numerical solution of Equation (5) as: 1, 2, , .
Note that, the scheme ( 11) is of O(h 2 ) accurate for the nu (12a) merical solution of (5), however, the scheme fails to compute at l = 1.We overcome this difficulty by using the following approximations: In order to obtain the compression spline sch the parabolic Equation (4), we replace by eme for (13) and Case 2: In this case, we consider the differential equation of the form This is a particular case of Equation (1), in which the function term   , u x t vation is absent.For the deri of the method, we follow the same id ].
eas given by Kadalbajoo and Patidar [14], and Mohanty et al. [15 We consider the ordinary differential equation which is a steady-state case of Equation (15).As in case 1, we seek S(x) as a solution of the above diffe equation rential This satisfies the interpolating conditions (7) and the continuity condition (8).
Solving the Equation ( 16) by the help of conditions (7), we obtain where x Similarly, replacing l by l + 1 in Equation ( 18), we can spline function in get the valid Differentiating Equation ( 18) with respect to "x" and using the continuity condition (8), we may obtain the spline in compression method for the approximate solution of Equation ( 16) as: where .Note that, the scheme (20) is of O(h 2 ) accurate erical solution of ( 16), however, the scheme (20) fails to compute at l = 1.We overcome this ty by us p hL  for the num difficul ing the approximations defined by ( 12) and we obtain In order to obtain the compression spline meth the parabolic Equation ( 15), we replace by and  2   Case 3: Finally we consider the most general problem (1), where both which is a steady-state case of (1).In this ca function se the spline This al tisfies the c so sa onditions (7) and (8).Solving the Equation (24) with the help of interpo tions (7), we obtain lating condi- where Replacing l by l + 1 in Equation (25), we can obtain the spline function k ] in Copyright © 2012 SciRes.
Using the continuity condition (8), from Equation (25) we obtain the difference scheme based on spline in compression for the approximate solution of Equation (23 O k h  accurate for the numerical solution of singularly perturbed parabolic partial differential Equations ( 4), ( 15) and ( 1), respectively and free from the terms   1 1 l x  , hence very easily computed for  , in the solution region  .

Stability Analysis
Now we discuss the stability analysis for the scheme (14).
In this case the exact solution where x l h a a h a Note that the compression spline scheme (27) is of for the approximate solution of the Equation However, this scheme fails when the coefficients we assume that there exists an error j j l l u   j l e U at each grid point (x l , t j ), then subtracting ( 14) from (29), we obtain the error equation   a x ,   b x and   f x contain singularities and the is to be determined at l = 1.We overcome this difficulty by modifying the scheme (27) in such a manner he ion reta s order and accuracy even in the vicinity of the singularity x = 0.
As discussed in case 1 and case 2, using the approximations (12) and neglecting high order terms, we obtain the following compression spline solution that t solut ins it scheme for the solution of To establish stability for the scheme (14), it is necessary to assume that the solution of the homogeneous part of the error Equation ( 30) is of the form , where  is in general complex, 1 i   ,  is real and we obtain the amplification factor parabolic Equation (1) in compact form: see (28).Note that the compression spline schemes ( 14), ( 22) . 2 4 32 and h   , it is easy to verify from (31) that 1   for all variable angle  .Hence the heme ( 14) is unconditionally stable.

Experimental Results
Numerical The exact solution is given by  

Fi Disc
The   is either small or large as comp h > 0 and k > 0. In Ta rors for the example 1 ared to the ble 1, we using the ra corresponding mesh sizes have reported the RMS er method discussed in case 1.In Table 2, we have given the RMS errors for the singularly perturbed parabolic Equation (33) in cylindrical and spherical polar coordinates using the method discussed in case 2. In Table 3, we have tabulated the RMS errors for the more gene l linear parabolic Equation (34) using the method discussed in case 3.All results confirmed that the proposed compression spline methods produce an oscillation-free solution for 0 f x t are continuous bounded functions defined in the semi-infinite region boundary conditions associated with Equation (1) are given by
and the time step size k > 0 in t-direction respectively, where N and J are positive integers.The mesh ratio parameter is given by

1
  everywhere in the solution region 0 1, t > 0. The technique used in this paper may be extended to derive other numerical methods, not necessarily limited to compression spline methods.