International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 302923 | 6 pages | https://doi.org/10.5402/2012/302923

Cubic Spline Method for 1D Wave Equation in Polar Coordinates

Academic Editor: P. B. Vasconcelos
Received10 Sep 2011
Accepted28 Sep 2011
Published20 Nov 2011

Abstract

Using nonpolynomial cubic spline approximation in space and finite difference in time direction, we discuss three-level implicit difference scheme of 𝑂(𝑘2+ℎ4) for the numerical solution of 1D wave equations in polar coordinates, where 𝑘>0 and ℎ>0 are grid sizes in time and space coordinates, respectively. The proposed method is applicable to problems with singularity. Stability theory of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

1. Introduction

We consider the 1D linear singular hyperbolic equation of the form𝜕2𝑢𝜕𝑡2=𝜕2𝑢𝜕𝑟2+𝛼𝑟𝜕𝑢𝜕𝑟+𝑓(𝑟,𝑡),0<𝑟<1,𝑡>0,𝛼=1,2(1) subject to the initial conditions𝑢(𝑟,0)=𝜙(𝑟),𝑢𝑡(𝑟,0)=𝜓(𝑟),0≤𝑟≤1(2) and with boundary conditions at 𝑟=0and𝑟=1of the form𝑢(0,𝑡)=𝑔0(𝑡),𝑢(1,𝑡)=𝑔1(𝑡),𝑡≥0,(3) where 𝑢=𝑢(𝑟,𝑡),𝑡 and 𝑟 are time and distance variable, respectively. For 𝛼=1 and 2, the equation above represent, 1D wave equation in cylindrical and spherical polar coordinates, respectively. We shall assume that the initial and boundary conditions are given with sufficient smoothness to maintain the order of accuracy of the difference scheme and spline functions under consideration.

In this paper, we are interested to discuss a new approximation based on cubic spline polynomial for the solution to singular hyperbolic equation (1). During last three decades, several numerical schemes for the solution of two-point boundary value problems and partial differential equations have been developed by many researchers. First Bickley [1] and Fyfe [2] have discussed the second-order accurate spline method for the solution of linear two-point boundary value problems. Jain and Aziz [3] have derived fourth-order cubic spline method for solving the nonlinear two-point boundary value problems with significant first derivative terms. Further, Khan and Aziz [4] have studied parametric cubic spline method to the solution of a system of second-order boundary value problem. In 1974, Raggett and Wilson [5] have used a cubic spline technique of lower order accuracy to solve the wave equation. Later, Fleck Jr [6] has proposed a cubic spline method for solving the wave equation of linear optics. In recent years, Mohanty [7], Gao and Chi [8], Rashidinia et al. [9, 10], Mohebbi and Dehghan [11], Ding and Zhang [12, 13], and Liu and Liu [14] have derived various numerical methods for solution of nonsingular 2D hyperbolic equations. A difference method of accuracy two in time and four in space for the solution of differential equation (1) has been studied by Mohanty [15] using finite difference method. A new discretization method of order four for the numerical solution of one-space dimensional second-order nonlinear hyperbolic equations has been studied by Mohanty et al. [16, 17]. Recently, Mohanty and Dahiya [18] have derived highly accurate cubic spline method for the solution of parabolic equations. In this paper, using nine grid points, we discuss a new three-level implicit cubic spline finite difference method of accuracy two in time and four in space for the solution of differential equation (1). To the authors’ knowledge, no cubic spline method of accuracy two in time and four in space for the solution of (1) has been discussed in the literature so far. In next section, we discuss the derivation of the proposed cubic spline method. It has been experienced in the past that the cubic spline solutions for the wave equation in polar coordinates usually deteriorate in the vicinity of the singularity. We overcome this difficulty by modifying the method in such a way that the solutions retain its order and accuracy everywhere in the vicinity of the singularity. In Section 3, we discuss the linear stability analysis of the proposed cubic spline method. In Section 4, we compare the computed results with the one obtained by the method discussed in [17]. Final remarks are given in Section 5.

2. The Method Based on Cubic Spline Approximation

The solution domain [0,1]×[𝑡>0] is divided into (𝑁+1)×𝐽 mesh with the spatial step size ℎ=1/(𝑁+1)in 𝑟-direction 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 𝜆=𝑘/ℎ>0. Grid points are defined by (𝑟𝑙,𝑡𝑗)=(ğ‘™â„Ž,𝑗𝑘),𝑙=0,1,2,…,𝐽. The notations 𝑢𝑗𝑙and𝑈𝑗𝑙 are used for the discrete approximation and the exact value of 𝑢(𝑟,𝑡) at the grid point (𝑟𝑙,𝑡𝑗), respectively.

At the grid point (𝑟𝑙,𝑡𝑗), we denoteğ‘ˆğ‘Žğ‘=ğœ•ğ‘Ž+ğ‘ğ‘ˆğœ•ğ‘Ÿğ‘Žğ‘™ğœ•ğ‘¡ğ‘ğ‘—,ğ‘“ğ‘Žğ‘=ğœ•ğ‘Ž+ğ‘ğ‘“ğœ•ğ‘Ÿğ‘Žğ‘™ğœ•ğ‘¡ğ‘ğ‘—.(4)

Let 𝑆𝑗(𝑟) be the cubic spline interpolating polynomial of the function 𝑢(𝑟,𝑡𝑗) between the grid points (𝑟𝑙−1,𝑡𝑗) and (𝑟𝑙,𝑡𝑗) and be given by𝑆𝑗𝑟(𝑟)=𝑙−𝑟3𝑀6â„Žğ‘—ğ‘™âˆ’1+𝑟−𝑟𝑙−13𝑀6â„Žğ‘—ğ‘™+𝑈𝑗𝑙−1−ℎ26𝑀𝑗𝑙−1î‚¶î‚€ğ‘Ÿğ‘™âˆ’ğ‘Ÿâ„Žî‚+î‚µğ‘ˆğ‘—ğ‘™âˆ’â„Ž26𝑀𝑗𝑙𝑟−𝑟𝑙−1ℎ,𝑟𝑙−1≤𝑟≤𝑟𝑙,𝑙=1,2,…,𝑁+1,𝑗=1,2,…,𝐽,(5) which satisfies at 𝑗th level the following properties:(i)𝑆𝑗(𝑟) coincides with a polynomial of degree three on each [𝑟𝑙−1,𝑟𝑙], 𝑙=1,2,…,𝑁+1, 𝑗=1,2,…𝐽,(ii)𝑆𝑗(𝑟)∈𝐶2[0,1], and(iii)𝑆𝑗(𝑟𝑙)=𝑈𝑗𝑙, 𝑙=0,1,2,…,𝑁+1, 𝑗=1,2,…,𝐽.

The derivatives of cubic spline function 𝑆𝑗(𝑟) are given by ğ‘†î…žğ‘—âˆ’î€·ğ‘Ÿ(𝑟)=𝑙−𝑟2𝑀2â„Žğ‘—ğ‘™âˆ’1+r−𝑟𝑙−12𝑀2â„Žğ‘—ğ‘™+𝑈𝑗𝑙−𝑈𝑗𝑙−1ℎ−ℎ6𝑀𝑗𝑙−𝑀𝑗𝑙−1,ğ‘†ğ‘—î…žî…žî€·ğ‘Ÿ(𝑟)=ğ‘™î€¸âˆ’ğ‘Ÿâ„Žğ‘€ğ‘—ğ‘™âˆ’1+𝑟−𝑟𝑙−1î€¸â„Žğ‘€ğ‘—ğ‘™,(6) where𝑀𝑗𝑙=ğ‘†ğ‘—î…žî…žî€·ğ‘Ÿğ‘™î€¸=𝑈𝑗𝑟𝑟𝑙=𝑈𝑗𝑡𝑡𝑙−𝛼𝑟𝑙𝑈𝑗𝑟𝑙−𝑓𝑗𝑙,𝑚𝑙=0,1,2,…,𝑁+1,𝑗=1,2,…,𝐽,(7)𝑗𝑙=ğ‘†î…žğ‘—î€·ğ‘Ÿğ‘™î€¸=𝑈𝑗𝑟𝑙=𝑈𝑗𝑙−𝑈𝑗𝑙−1ℎ+ℎ6𝑀𝑗𝑙−1+2𝑀𝑗𝑙,𝑟𝑙−1≤𝑟≤𝑟𝑙,(8) and replacing ℎ by “−ℎ,” we get𝑚𝑗𝑙=ğ‘†î…žğ‘—î€·ğ‘Ÿğ‘™î€¸=𝑈𝑗𝑟𝑙=𝑈𝑗𝑙+1âˆ’ğ‘ˆğ‘—ğ‘™â„Žâˆ’â„Ž6𝑀𝑗𝑙+1+2𝑀𝑗𝑙,𝑟𝑙−1≤𝑟≤𝑟𝑙.(9) Combining (8) and (9), we obtain 𝑚𝑗𝑙=ğ‘†î…žğ‘—î€·ğ‘Ÿğ‘™î€¸=𝑈𝑗𝑟𝑙=𝑈𝑗𝑙+1−𝑈𝑗𝑙−1−ℎ2â„Žî€ºğ‘€12𝑗𝑙+1−𝑀𝑗𝑙−1.(10) Further, from (8), we have𝑚𝑗𝑙+1=ğ‘†î…žğ‘—î€·ğ‘Ÿğ‘™+1=𝑈𝑗𝑟𝑙+1=𝑈𝑗𝑙+1âˆ’ğ‘ˆğ‘—ğ‘™â„Ž+ℎ6𝑀𝑗𝑙+2𝑀𝑗𝑙+1,(11) and, from (9), we have𝑚𝑗𝑙−1=ğ‘†î…žğ‘—î€·ğ‘Ÿğ‘™âˆ’1=𝑈𝑗𝑟𝑙−1=𝑈𝑗𝑙−𝑈𝑗𝑙−1ℎ−ℎ6𝑀𝑗𝑙+2𝑀𝑗𝑙−1.(12) We consider the following approximations:𝑈𝑗𝑡𝑙=𝑈𝑙𝑗+1−𝑈𝑙𝑗−12𝑘=𝑈𝑡𝑗𝑙𝑘+𝑂2,𝑈𝑗𝑡𝑙+1=𝑈𝑗+1𝑙+1−𝑈𝑗−1𝑙+12𝑘=𝑈𝑗𝑡𝑙+1𝑘+𝑂2+𝑘2ℎ,𝑈𝑗𝑡𝑙−1=𝑈𝑗+1𝑙−1−𝑈𝑗−1𝑙−12𝑘=𝑈𝑗𝑡𝑙−1𝑘+𝑂2−𝑘2ℎ,𝑈𝑗𝑡𝑡𝑙=𝑈𝑙𝑗+1−2𝑈𝑗𝑙+𝑈𝑙𝑗−1𝑘2=𝑈𝑗𝑡𝑡𝑙𝑘+𝑂2,𝑈𝑗𝑡𝑡𝑙+1=𝑈𝑗+1𝑙+1−2𝑈𝑗𝑙+1+𝑈𝑗−1𝑙+1𝑘2=𝑈𝑗𝑡𝑡𝑙+1𝑘+𝑂2+𝑘2ℎ,𝑈𝑗𝑡𝑡𝑙−1=𝑈𝑗+1𝑙−1−2𝑈𝑗𝑙−1+𝑈𝑗−1𝑙−1𝑘2=𝑈𝑗𝑡𝑡𝑙−1𝑘+𝑂2−𝑘2ℎ,𝑈𝑗𝑟𝑙=𝑈𝑗𝑙+1−𝑈𝑗𝑙−12ℎ=𝑈𝑗𝑟𝑙+ℎ26𝑈30𝑘+𝑂2+ℎ4,𝑈𝑗𝑟𝑙+1=3𝑈𝑗𝑙+1−4𝑈𝑗𝑙+𝑈𝑗𝑙−12ℎ=𝑈𝑗𝑟𝑙+1−ℎ23𝑈30𝑘+𝑂2+𝑘2ℎ,𝑈𝑗𝑟𝑙−1=−3𝑈𝑗𝑙−1+4𝑈𝑗𝑙−𝑈𝑗𝑙+12ℎ=𝑈𝑗𝑟𝑙−1−ℎ23𝑈30𝑘+𝑂2−𝑘2ℎ.(13) Since the derivative values of 𝑆𝑗(𝑟) defined by (7), (10), (11), and (12) are not known at each grid point (𝑟𝑙,𝑡𝑗), we use the following approximations for the derivatives of 𝑆𝑗(𝑟).

Let 𝑀𝑗𝑙=𝑈𝑗𝑡𝑡𝑙−𝛼𝑟𝑙𝑈𝑗𝑟𝑙−𝑓𝑗𝑙,𝑀𝑗𝑙+1=𝑈𝑗𝑡𝑡𝑙+1−𝛼𝑟𝑙+1𝑈𝑗𝑟𝑙+1−𝑓𝑗𝑙+1,𝑀𝑗𝑙−1=𝑈𝑗𝑡𝑡𝑙−1−𝛼𝑟𝑙−1𝑈𝑗𝑟𝑙−1−𝑓𝑗𝑙−1,𝑚𝑗𝑙=𝑈𝑗𝑙+1−𝑈𝑗𝑙−1−ℎ2ℎ12𝑀𝑗𝑙+1−𝑀𝑗𝑙−1,𝑚𝑗𝑙+1=𝑈𝑗𝑙+1âˆ’ğ‘ˆğ‘—ğ‘™â„Ž+ℎ6𝑀𝑗𝑙+2𝑀𝑗𝑙+1,𝑚𝑗𝑙−1=𝑈𝑗𝑙−𝑈𝑗𝑙−1ℎ−ℎ6𝑀𝑗𝑙+2𝑀𝑗𝑙−1,(14) where 𝑚𝑗𝑙 and 𝑀𝑗𝑙 are approximations to 𝑢𝑗𝑟𝑙 and 𝑢𝑗𝑟𝑟𝑙, respectively.

We may rewrite the differential equation (1) as𝜕2𝑢𝜕𝑟2=𝜕2𝑢𝜕𝑡2−𝛼𝑟𝜕𝑢𝜕𝑟−𝑓(𝑟,𝑡)≡𝜙(𝑟,𝑡)(say).(15)

Let us denote 𝜙𝑗𝑙=𝜙(𝑟𝑙,𝑡𝑗).

A fourth-order method (see [3]) based on cubic spline approximations for the differential equation (15) may be written as𝑈𝑗𝑙+1−2𝑈𝑗𝑙+𝑈𝑗𝑙−1=ℎ2𝜙12𝑗𝑙+1+𝜙𝑗𝑙−1𝜙+10𝑗𝑙,(16) where𝜙𝑗𝑙+1=𝑈𝑗𝑡𝑡𝑙+1−𝛼𝑟𝑙+1𝑚𝑗𝑙+1−𝑓𝑗𝑙+1,𝜙𝑗𝑙−1=𝑈𝑗𝑡𝑡𝑙−1−𝛼𝑟𝑙−1𝑚𝑗𝑙−1−𝑓𝑗𝑙−1,𝜙𝑗𝑙=𝑈𝑗𝑡𝑡𝑙−𝛼𝑟𝑙𝑚𝑗𝑙−𝑓𝑗𝑙.(17)

Multiplying throughout by 6𝜆2 and by the help of the approximations (17), from (16), we obtain a cubic spline finite difference method with accuracy of 𝑂(𝑘2+ℎ4) for the solution of differential equation (1) as6𝜆2𝑈𝑗𝑙+1−2𝑈𝑗𝑙+𝑈𝑗𝑙−1=𝑘22𝑈𝑗𝑡𝑡𝑙+1+𝑈𝑗𝑡𝑡𝑙−1+10𝑈𝑗𝑡𝑡𝑙−𝑘22𝛼𝑟𝑙+1𝑚𝑗𝑙+1+𝛼𝑟𝑙−1𝑚𝑗𝑙−1𝛼+10𝑟𝑙𝑚𝑗𝑙−𝑘22𝑓𝑗𝑙+1+𝑓𝑗𝑙−1+10𝑓𝑗𝑙.(18)

Note that the method (18) is of 𝑂(𝑘2+ℎ4) for the numerical solution of (1). However, the method fails to compute at 𝑙=1. We modify the method in such a manner that it retains its order and accuracy in the vicinity of the singularity.

We use the following approximations:1𝑟𝑙+1=1ğ‘Ÿğ‘™âˆ’â„Žğ‘Ÿ2𝑙+ℎ2𝑟3ğ‘™î€·â„Žâˆ’ğ‘‚3,1𝑟𝑙−1=1𝑟𝑙+â„Žğ‘Ÿ2𝑙+ℎ2𝑟3ğ‘™î€·â„Ž+𝑂3,𝑓𝑗𝑙+1=𝑓𝑗𝑙+â„Žğ‘“10+ℎ22𝑓20ℎ+𝑂3,𝑓𝑗𝑙−1=ğ‘“ğ‘—ğ‘™âˆ’â„Žğ‘“10+ℎ22𝑓20î€·â„Žâˆ’ğ‘‚3,𝛼𝑟𝑙+1𝑈𝑗𝑟𝑙+1+𝑓𝑗𝑙+1=1𝛼2â„Žğ‘Ÿğ‘™+12𝜇𝑟𝛿𝑟+2𝛿2𝑟𝑈𝑗𝑙+𝑓𝑗𝑙+1,𝛼𝑟𝑙−1𝑈𝑗𝑟𝑙−1+𝑓𝑗𝑙−1=1𝛼2â„Žğ‘Ÿğ‘™âˆ’12𝜇𝑟𝛿𝑟−2𝛿2𝑟𝑈𝑗𝑙+𝑓𝑗𝑙−1,𝛼𝑟𝑙𝑈𝑗𝑟𝑙+𝑓𝑗𝑙=1𝛼2â„Žğ‘Ÿğ‘™î€·2𝜇𝑟𝛿𝑟𝑈𝑗𝑙+𝑓𝑗𝑙.(19) where 𝜇𝑟𝑢𝑙=(1/2)(𝑢1+(1/2)+𝑢1−(1/2)) and 𝛿𝑟𝑢𝑙=(𝑢1+(1/2)−𝑢1−(1/2)). Substituting the approximations (19) into (18) and neglecting higher order terms, we get12+𝛿2𝑟𝛿2𝑡𝑈𝑗𝑙−12𝜆2𝛿2𝑟𝑈𝑗𝑙=𝜆2ℎ2𝛼12𝑟𝑙+2ℎ2𝛼𝑟3𝑙2𝜇𝑟𝛿𝑟𝑈𝑗𝑙+𝜆2ℎ2𝛼(𝛼−1)𝑟2𝑙𝛿2ğ‘Ÿğ‘ˆğ‘—ğ‘™âˆ’â„Ž2𝛼𝑟2𝑙𝛿2ğ‘¡ğ‘ˆğ‘—ğ‘™âˆ’â„Žğ›¼2𝑟𝑙𝛿2𝑡2𝜇𝑟𝛿𝑟𝑈𝑗𝑙+𝑘212𝑓00+ℎ2𝑓20+𝛼𝑟𝑙𝑓10−𝑓00.(20)

Note that the cubic spline method (20) is of 𝑂(𝑘2+ℎ4) for the numerical solution of differential equation (1), which is free from the terms 1/𝑟𝑙±1 hence can be computed for 𝑙=1(1)𝑁,𝑗=0,1,2,….

3. Stability Analysis

Now we discuss the stability analysis for the scheme (20). For stability, we consider the homogeneous part of the scheme (20), which can be written as 𝑅0+1𝛿122𝑟+𝑅12𝜇𝑟𝛿𝑟𝛿2𝑡𝑈𝑗𝑙=𝜆2𝑅2𝛿2𝑟+𝑅32𝜇𝑟𝛿𝑟𝑈𝑗𝑙,(21) where 𝑅0ℎ=1+2𝛼12𝑟2,𝑅1=â„Žğ›¼2𝑟,𝑅2ℎ=1+212𝛼(𝛼−1)𝑟2,𝑅3=𝑅1+ℎ3242𝛼𝑟3.(22)

It is difficult to find the stability region for the scheme (21). In order to obtain a valid stability region, we may modify the scheme (21) (see [19]) as𝑅0+1𝑅122𝛿2𝑟+𝑅32𝜇𝑟𝛿𝑟𝛿2𝑡𝑈𝑗𝑙=𝜆2𝑅2𝛿2𝑟+𝑅32𝜇𝑟𝛿𝑟𝑈𝑗𝑙.(23)

The additional terms added in (23) are of high order and do not affect the accuracy of the scheme.

Put 𝑈𝑗𝑙=𝐴𝑙𝜉𝑗𝑒𝑖𝛽𝑙=𝐴𝑙𝑒𝑖𝜙𝑗𝑒𝑖𝛽𝑙 (where 𝜉=𝑒𝑖𝜙 such that |𝜉|=1) in the modified scheme (23), we get𝜉−2+𝜉−1=−4sin2𝜙2=𝜆2𝑅2𝐴+𝐴−1cos𝛽−2+𝑖𝐴−𝐴−1sin𝛽+𝑅3𝐴−𝐴−1cos𝛽+𝑖𝐴+𝐴−1sin𝛽𝑅0+1𝑅122𝐴+𝐴−1cos𝛽−2+𝑖𝐴−𝐴−1sin𝛽+𝑅3𝐴−𝐴−1cos𝛽+𝑖𝐴+𝐴−1,sin𝛽(24)where 𝐴 is a nonzero real parameter to be determined. Left-hand side of (24) is a real quantity. Thus, the imaginary part of right-hand side of (24) must be zero.

Thus, we obtain𝑅2𝐴−𝐴−1+𝑅3𝐴+𝐴−1⟹𝑅=02+𝑅3𝑅𝐴−2−𝑅3𝐴−1=0⟹𝐴=𝑅2−𝑅3𝑅2+𝑅3,𝐴−1=îƒŽğ‘…2+𝑅3𝑅2−𝑅3(25) provided𝑅2±𝑅3>0,thatis,−𝑅2<𝑅3<𝑅2.

Hence, 𝐴+𝐴−1=2𝑅2/𝑅22−𝑅23and𝐴−𝐴−1=−2𝑅3/𝑅22−𝑅23.

Substituting the values of 𝐴,𝐴−1,𝐴+𝐴−1,and𝐴−𝐴−1in (24) and (25), we obtain−4sin2𝜙2=𝜆22𝑅22−𝑅23cos𝛽−2𝑅2𝑅0+(1/6)𝑅22−𝑅23cos𝛽−𝑅2=2𝜆2𝑅22−𝑅23cos𝛽−𝑅2𝑅0+(1/6)𝑅22−𝑅23cos𝛽−𝑅2⟹sin2𝜙2=𝜆2𝑅2−𝑅22−𝑅23cos𝛽2𝑅0+(1/3)𝑅22−𝑅23cos𝛽−𝑅2=𝜆2𝑅2+𝑅22−𝑅232sin2𝛽/2−12𝑅0𝑅−(1/3)2+𝑅22−𝑅232sin2≡𝑁𝛽/2−1𝐷.(26)

Since 0≤sin2𝜙/2≤1, the method (23) is stable as long as 𝑁≤𝐷, which is true if 𝑁≤Max(𝑁)≤Min(𝐷)≤𝐷.

Hence, the required stability condition is𝜆Max2𝑅2+𝑅22−𝑅232sin2𝛽2−1≤Min2𝑅0−13𝑅2+𝑅22−𝑅232sin2𝛽2−1⟹𝜆2𝑅2+𝑅22−𝑅23≤2𝑅0−13𝑅2−𝑅22−𝑅230<𝜆2≤2𝑅0𝑅−(1/3)2−𝑅22−𝑅23𝑅2+𝑅22−𝑅23,(27) which is the required stability interval for the scheme (23).

4. Numerical Results

A difference method of 𝑂(𝑘2+ℎ2), for the differential equation (1) may be written as𝑢𝑗𝑡𝑡𝑙=𝑢𝑗𝑟𝑟𝑙+𝛼𝑟𝑢𝑗𝑟𝑙𝑟+𝑓𝑙,𝑡𝑗,𝑙=1(1)𝑁,𝑗=0,1,2,….(28)

Note that the proposed cubic spline method (23) and the difference method (28) for second-order hyperbolic equation (1) are three-level schemes. The value of u at 𝑡=0 is known from the initial condition. To start any computation, it is necessary to know the numerical value of u of required accuracy at 𝑡=𝑘. In this section, we discuss an explicit scheme of 𝑂(𝑘2) for u at first time level, that is, at 𝑡=𝑘, in order to solve the differential equation (1) using the proposed method (23) and second-order method (28).

Since the values of 𝑢 and 𝑢𝑡 are known explicitly at 𝑡=0, this implies all their successive tangential derivatives are known at 𝑡=0, that is, the values of 𝑢,𝑢𝑟,𝑢𝑟𝑟,…,𝑢𝑡,𝑢𝑡𝑟,…, and so forth, are known at 𝑡=0.

An approximation for 𝑢 of 𝑂(𝑘2) at 𝑡=𝑘 may be written as𝑢1𝑙=𝑢0𝑙+𝑘𝑢0𝑡𝑙+𝑘22𝑢𝑡𝑡0𝑙𝑘+𝑂3.(29)

From (1), we have𝑢𝑡𝑡0𝑙=𝑢𝑟𝑟+𝛼𝑟𝑢𝑟+𝑓(𝑟,𝑡)0𝑙.(30)

Thus, using the initial values and their successive tangential derivative values, from (30), we can obtain the value of (𝑢𝑡𝑡)0l, and, then ultimately, from (29), we can compute the value of 𝑢 at first time level, that is, at 𝑡=𝑘.

We solve the differential equation (1) in the region 0<𝑟<1,𝑡>0, whose exact solution is given by 𝑢(𝑟,𝑡)=cosh(𝑟)⋅sin(𝑡). The maximum absolute errors at 𝑡=5.0 are tabulated in Table 1 for various values of 𝜆=𝑘/ℎ=0.8 and in Table 2 for 𝛾=𝑘/ℎ2=4.0. We have compared the numerical results of the proposed method with the results obtained by using the method discussed in [17] in terms of accuracy. All computations were performed using double precision arithmetic.


h Proposed 𝑂 ( 𝑘 2 + ℎ 4 ) method 𝑂 ( 𝑘 2 + ℎ 2 ) method
𝛼 = 1 𝛼 = 2 𝛼 = 1 𝛼 = 2

1/160.6222 (−4)0.3981 (−4)0.2094 (−3)0.2760 (−3)
1/320.1593 (−4)0.1057 (−4)0.5404 (−4)0.6849 (−4)
1/640.3997 (−5)0.2699 (−5)0.1365 (−4)0.1705 (−4)
1/1280.9944 (−6)0.6797 (−6)0.3412 (−5)0.4257 (−5)


hProposed 𝑂 ( 𝑘 2 + ℎ 4 ) method 𝑂 ( 𝑘 4 + ℎ 4 ) method
discussed in [17]
𝛼 = 1 𝛼 = 2 𝛼 = 1 𝛼 = 2

1/100.3227 (−4)0.1658 (−4)0.6280 (−4)0.5558 (−4)
1/200.1967 (−5)0.1019 (−5)0.3884 (−5)0.3378 (−5)
1/400.1174 (−6)0.6282 (−7)0.2322 (−6)0.2084 (−6)
1/800.6935 (−8)0.3897 (−8)0.1408 (−7)0.1266 (−7)

A relation between the exact solution ğ‘¢ğ‘’ğ‘¥ğ‘Žğ‘ğ‘¡ and the approximate numerical solution 𝑢(ℎ) as given in the following equation:𝑢exact=𝑢(ℎ)+ğ´â„Žğ‘+⋯higher-orderterms,(31) where ℎ is the measure of the mesh discretization, 𝐴 is a constant, and 𝑝 is the order (rate) of convergence. If the meshes to be considered are sufficiently refined, the higher-order terms can be neglected. Then, the maximum absolute errors ğ¸â„Ž can be approximated as ğ¸â„Ž||𝑢=Maxexact||−𝑢(ℎ)â‰…ğ´â„Žğ‘.(32)

Taking the logarithm of both sides of (32), we obtain𝐸logℎ=log(𝐴)+𝑝log(ℎ).(33)

For two different refined mesh spacing ℎ1 and ℎ2, we have the following two relations𝐸logℎ1ℎ=log(𝐴)+𝑝log1,(34a)𝐸logℎ2ℎ=log(𝐴)+𝑝log2.(34b)

Subtracting (34b) from (34a), we obtain the order (rate) of convergence 𝐸𝑝=logℎ1𝐸−logℎ2ℎlog1ℎ−log2,(35) where ğ¸â„Ž1 and ğ¸â„Ž2 are maximum absolute errors for two uniform mesh widths ℎ1 and ℎ2, respectively. For computation of order of convergence of the proposed method, we have considered ℎ1=1/40 and ℎ2=1/80 in Table 2, and we found the order of convergence of the proposed method for 𝛼=1is 4.08 and for 𝛼=2 is 4.01.

5. Final Remarks

Available numerical methods based on spline approximations for the numerical solution of 1D wave equation in polar coordinates are of 𝑂(𝑘2+ℎ2) accurate, which require nine grid points. In this paper, using the same number of grid points, we have discussed a new stable three-level implicit cubic spline finite difference method of 𝑂(𝑘2+ℎ4) accuracy for the solution of wave equation in polar coordinates. For a fixed parameter 𝛾=𝑘/ℎ2, the proposed method behaves like a fourth-order method, which is exhibited from the computed results.

Acknowledgment

The authors thank the anonymous reviewers for their constructive suggestions, which substantially improved the standard of the paper.

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Copyright © 2012 R. K. Mohanty et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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