International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

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

R. K. Mohanty, Rajive Kumar, Vijay Dahiya, "Cubic Spline Method for 1D Wave Equation in Polar Coordinates", International Scholarly Research Notices, vol. 2012, Article ID 302923, 6 pages, 2012. 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+4) for the numerical solution of 1D wave equations in polar coordinates, where 𝑘>0 and >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),0𝑟1(2) and with boundary conditions at 𝑟=0and𝑟=1of the form𝑢(0,𝑡)=𝑔0(𝑡),𝑢(1,𝑡)=𝑔1(𝑡),𝑡0,(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 =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/>0. Grid points are defined by (𝑟𝑙,𝑡𝑗)=(𝑙,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 denote𝑈𝑎𝑏=𝜕𝑎+𝑏𝑈𝜕𝑟𝑎𝑙𝜕𝑡𝑏𝑗,𝑓𝑎𝑏=𝜕𝑎+𝑏𝑓𝜕𝑟𝑎𝑙𝜕𝑡𝑏𝑗.(4)

Let 𝑆𝑗(𝑟) be the cubic spline interpolating polynomial of the function 𝑢(𝑟,𝑡𝑗) between the grid points (𝑟𝑙1,𝑡𝑗) and (𝑟𝑙,𝑡𝑗) and be given by𝑆𝑗𝑟(𝑟)=𝑙𝑟3𝑀6𝑗𝑙1+𝑟𝑟𝑙13𝑀6𝑗𝑙+𝑈𝑗𝑙126𝑀𝑗𝑙1𝑟𝑙𝑟+𝑈𝑗𝑙26𝑀𝑗𝑙𝑟𝑟𝑙1,𝑟𝑙1𝑟𝑟𝑙,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 [𝑟𝑙1,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 𝑆𝑗𝑟(𝑟)=𝑙𝑟2𝑀2𝑗𝑙1+r𝑟𝑙12𝑀2𝑗𝑙+𝑈𝑗𝑙𝑈𝑗𝑙16𝑀𝑗𝑙𝑀𝑗𝑙1,𝑆𝑗𝑟(𝑟)=𝑙𝑟𝑀𝑗𝑙1+𝑟𝑟𝑙1𝑀𝑗𝑙,(6) where𝑀𝑗𝑙=𝑆𝑗𝑟𝑙=𝑈𝑗𝑟𝑟𝑙=𝑈𝑗𝑡𝑡𝑙𝛼𝑟𝑙𝑈𝑗𝑟𝑙𝑓𝑗𝑙,𝑚𝑙=0,1,2,,𝑁+1,𝑗=1,2,,𝐽,(7)𝑗𝑙=𝑆𝑗𝑟𝑙=𝑈𝑗𝑟𝑙=𝑈𝑗𝑙𝑈𝑗𝑙1+6𝑀𝑗𝑙1+2𝑀𝑗𝑙,𝑟𝑙1𝑟𝑟𝑙,(8) and replacing by “,” we get𝑚𝑗𝑙=𝑆𝑗𝑟𝑙=𝑈𝑗𝑟𝑙=𝑈𝑗𝑙+1𝑈𝑗𝑙6𝑀𝑗𝑙+1+2𝑀𝑗𝑙,𝑟𝑙1𝑟𝑟𝑙.(9) Combining (8) and (9), we obtain 𝑚𝑗𝑙=𝑆𝑗𝑟𝑙=𝑈𝑗𝑟𝑙=𝑈𝑗𝑙+1𝑈𝑗𝑙12𝑀12𝑗𝑙+1𝑀𝑗𝑙1.(10) Further, from (8), we have𝑚𝑗𝑙+1=𝑆𝑗𝑟𝑙+1=𝑈𝑗𝑟𝑙+1=𝑈𝑗𝑙+1𝑈𝑗𝑙+6𝑀𝑗𝑙+2𝑀𝑗𝑙+1,(11) and, from (9), we have𝑚𝑗𝑙1=𝑆𝑗𝑟𝑙1=𝑈𝑗𝑟𝑙1=𝑈𝑗𝑙𝑈𝑗𝑙16𝑀𝑗𝑙+2𝑀𝑗𝑙1.(12) We consider the following approximations:𝑈𝑗𝑡𝑙=𝑈𝑙𝑗+1𝑈𝑙𝑗12𝑘=𝑈𝑡𝑗𝑙𝑘+𝑂2,𝑈𝑗𝑡𝑙+1=𝑈𝑗+1𝑙+1𝑈𝑗1𝑙+12𝑘=𝑈𝑗𝑡𝑙+1𝑘+𝑂2+𝑘2,𝑈𝑗𝑡𝑙1=𝑈𝑗+1𝑙1𝑈𝑗1𝑙12𝑘=𝑈𝑗𝑡𝑙1𝑘+𝑂2𝑘2,𝑈𝑗𝑡𝑡𝑙=𝑈𝑙𝑗+12𝑈𝑗𝑙+𝑈𝑙𝑗1𝑘2=𝑈𝑗𝑡𝑡𝑙𝑘+𝑂2,𝑈𝑗𝑡𝑡𝑙+1=𝑈𝑗+1𝑙+12𝑈𝑗𝑙+1+𝑈𝑗1𝑙+1𝑘2=𝑈𝑗𝑡𝑡𝑙+1𝑘+𝑂2+𝑘2,𝑈𝑗𝑡𝑡𝑙1=𝑈𝑗+1𝑙12𝑈𝑗𝑙1+𝑈𝑗1𝑙1𝑘2=𝑈𝑗𝑡𝑡𝑙1𝑘+𝑂2𝑘2,𝑈𝑗𝑟𝑙=𝑈𝑗𝑙+1𝑈𝑗𝑙12=𝑈𝑗𝑟𝑙+26𝑈30𝑘+𝑂2+4,𝑈𝑗𝑟𝑙+1=3𝑈𝑗𝑙+14𝑈𝑗𝑙+𝑈𝑗𝑙12=𝑈𝑗𝑟𝑙+123𝑈30𝑘+𝑂2+𝑘2,𝑈𝑗𝑟𝑙1=3𝑈𝑗𝑙1+4𝑈𝑗𝑙𝑈𝑗𝑙+12=𝑈𝑗𝑟𝑙123𝑈30𝑘+𝑂2𝑘2.(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=𝑈𝑗𝑡𝑡𝑙𝛼𝑟𝑙𝑈𝑗𝑟𝑙𝑓𝑗𝑙,𝑀𝑗𝑙+1=𝑈𝑗𝑡𝑡𝑙+1𝛼𝑟𝑙+1𝑈𝑗𝑟𝑙+1𝑓𝑗𝑙+1,𝑀𝑗𝑙1=𝑈𝑗𝑡𝑡𝑙1𝛼𝑟𝑙1𝑈𝑗𝑟𝑙1𝑓𝑗𝑙1,𝑚𝑗𝑙=𝑈𝑗𝑙+1𝑈𝑗𝑙1212𝑀𝑗𝑙+1𝑀𝑗𝑙1,𝑚𝑗𝑙+1=𝑈𝑗𝑙+1𝑈𝑗𝑙+6𝑀𝑗𝑙+2𝑀𝑗𝑙+1,𝑚𝑗𝑙1=𝑈𝑗𝑙𝑈𝑗𝑙16𝑀𝑗𝑙+2𝑀𝑗𝑙1,(14) where 𝑚𝑗𝑙 and 𝑀𝑗𝑙 are approximations to 𝑢𝑗𝑟𝑙 and 𝑢𝑗𝑟𝑟𝑙, respectively.

We may rewrite the differential equation (1) as𝜕2𝑢𝜕𝑟2=𝜕2𝑢𝜕𝑡2𝛼𝑟𝜕𝑢𝜕𝑟𝑓(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+12𝑈𝑗𝑙+𝑈𝑗𝑙1=2𝜙12𝑗𝑙+1+𝜙𝑗𝑙1𝜙+10𝑗𝑙,(16) where𝜙𝑗𝑙+1=𝑈𝑗𝑡𝑡𝑙+1𝛼𝑟𝑙+1𝑚𝑗𝑙+1𝑓𝑗𝑙+1,𝜙𝑗𝑙1=𝑈𝑗𝑡𝑡𝑙1𝛼𝑟𝑙1𝑚𝑗𝑙1𝑓𝑗𝑙1,𝜙𝑗𝑙=𝑈𝑗𝑡𝑡𝑙𝛼𝑟𝑙𝑚𝑗𝑙𝑓𝑗𝑙.(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+4) for the solution of differential equation (1) as6𝜆2𝑈𝑗𝑙+12𝑈𝑗𝑙+𝑈𝑗𝑙1=𝑘22𝑈𝑗𝑡𝑡𝑙+1+𝑈𝑗𝑡𝑡𝑙1+10𝑈𝑗𝑡𝑡𝑙𝑘22𝛼𝑟𝑙+1𝑚𝑗𝑙+1+𝛼𝑟𝑙1𝑚𝑗𝑙1𝛼+10𝑟𝑙𝑚𝑗𝑙𝑘22𝑓𝑗𝑙+1+𝑓𝑗𝑙1+10𝑓𝑗𝑙.(18)

Note that the method (18) is of 𝑂(𝑘2+4) 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=1𝑟𝑙𝑟2𝑙+2𝑟3𝑙𝑂3,1𝑟𝑙1=1𝑟𝑙+𝑟2𝑙+2𝑟3𝑙+𝑂3,𝑓𝑗𝑙+1=𝑓𝑗𝑙+𝑓10+22𝑓20+𝑂3,𝑓𝑗𝑙1=𝑓𝑗𝑙𝑓10+22𝑓20𝑂3,𝛼𝑟𝑙+1𝑈𝑗𝑟𝑙+1+𝑓𝑗𝑙+1=1𝛼2𝑟𝑙+12𝜇𝑟𝛿𝑟+2𝛿2𝑟𝑈𝑗𝑙+𝑓𝑗𝑙+1,𝛼𝑟𝑙1𝑈𝑗𝑟𝑙1+𝑓𝑗𝑙1=1𝛼2𝑟𝑙12𝜇𝑟𝛿𝑟2𝛿2𝑟𝑈𝑗𝑙+𝑓𝑗𝑙1,𝛼𝑟𝑙𝑈𝑗𝑟𝑙+𝑓𝑗𝑙=1𝛼2𝑟𝑙2𝜇𝑟𝛿𝑟𝑈𝑗𝑙+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 get12+𝛿2𝑟𝛿2𝑡𝑈𝑗𝑙12𝜆2𝛿2𝑟𝑈𝑗𝑙=𝜆22𝛼12𝑟𝑙+22𝛼𝑟3𝑙2𝜇𝑟𝛿𝑟𝑈𝑗𝑙+𝜆22𝛼(𝛼1)𝑟2𝑙𝛿2𝑟𝑈𝑗𝑙2𝛼𝑟2𝑙𝛿2𝑡𝑈𝑗𝑙𝛼2𝑟𝑙𝛿2𝑡2𝜇𝑟𝛿𝑟𝑈𝑗𝑙+𝑘212𝑓00+2𝑓20+𝛼𝑟𝑙𝑓10𝑓00.(20)

Note that the cubic spline method (20) is of 𝑂(𝑘2+4) 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+1𝛿122𝑟+𝑅12𝜇𝑟𝛿𝑟𝛿2𝑡𝑈𝑗𝑙=𝜆2𝑅2𝛿2𝑟+𝑅32𝜇𝑟𝛿𝑟𝑈𝑗𝑙,(21) where 𝑅0=1+2𝛼12𝑟2,𝑅1=𝛼2𝑟,𝑅2=1+212𝛼(𝛼1)𝑟2,𝑅3=𝑅1+3242𝛼𝑟3.(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]) as𝑅0+1𝑅122𝛿2𝑟+𝑅32𝜇𝑟𝛿𝑟𝛿2𝑡𝑈𝑗𝑙=𝜆2𝑅2𝛿2𝑟+𝑅32𝜇𝑟𝛿𝑟𝑈𝑗𝑙.(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 get𝜉2+𝜉1=4sin2𝜙2=𝜆2𝑅2𝐴+𝐴1cos𝛽2+𝑖𝐴𝐴1sin𝛽+𝑅3𝐴𝐴1cos𝛽+𝑖𝐴+𝐴1sin𝛽𝑅0+1𝑅122𝐴+𝐴1cos𝛽2+𝑖𝐴𝐴1sin𝛽+𝑅3𝐴𝐴1cos𝛽+𝑖𝐴+𝐴1,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 obtain𝑅2𝐴𝐴1+𝑅3𝐴+𝐴1𝑅=02+𝑅3𝑅𝐴2𝑅3𝐴1=0𝐴=𝑅2𝑅3𝑅2+𝑅3,𝐴1=𝑅2+𝑅3𝑅2𝑅3(25) provided𝑅2±𝑅3>0,thatis,𝑅2<𝑅3<𝑅2.

Hence, 𝐴+𝐴1=2𝑅2/𝑅22𝑅23and𝐴𝐴1=2𝑅3/𝑅22𝑅23.

Substituting the values of 𝐴,𝐴1,𝐴+𝐴1,and𝐴𝐴1in (24) and (25), we obtain4sin2𝜙2=𝜆22𝑅22𝑅23cos𝛽2𝑅2𝑅0+(1/6)𝑅22𝑅23cos𝛽𝑅2=2𝜆2𝑅22𝑅23cos𝛽𝑅2𝑅0+(1/6)𝑅22𝑅23cos𝛽𝑅2sin2𝜙2=𝜆2𝑅2𝑅22𝑅23cos𝛽2𝑅0+(1/3)𝑅22𝑅23cos𝛽𝑅2=𝜆2𝑅2+𝑅22𝑅232sin2𝛽/212𝑅0𝑅(1/3)2+𝑅22𝑅232sin2𝑁𝛽/21𝐷.(26)

Since 0sin2𝜙/21, the method (23) is stable as long as 𝑁𝐷, which is true if 𝑁Max(𝑁)Min(𝐷)𝐷.

Hence, the required stability condition is𝜆Max2𝑅2+𝑅22𝑅232sin2𝛽21Min2𝑅013𝑅2+𝑅22𝑅232sin2𝛽21𝜆2𝑅2+𝑅22𝑅232𝑅013𝑅2𝑅22𝑅230<𝜆22𝑅0𝑅(1/3)2𝑅22𝑅23𝑅2+𝑅22𝑅23,(27) which is the required stability interval for the scheme (23).

4. Numerical Results

A difference method of 𝑂(𝑘2+2), for the differential equation (1) may be written as𝑢𝑗𝑡𝑡𝑙=𝑢𝑗𝑟𝑟𝑙+𝛼𝑟𝑢𝑗𝑟𝑙𝑟+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+𝑘22𝑢𝑡𝑡0𝑙𝑘+𝑂3.(29)

From (1), we have𝑢𝑡𝑡0𝑙=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/=0.8 and in Table 2 for 𝛾=𝑘/2=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 + 4 ) method 𝑂 ( 𝑘 2 + 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 + 4 ) method 𝑂 ( 𝑘 4 + 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 𝑢𝑒𝑥𝑎𝑐𝑡 and the approximate numerical solution 𝑢() as given in the following equation:𝑢exact=𝑢()+𝐴𝑝+higher-orderterms,(31) where 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 𝐸 can be approximated as 𝐸||𝑢=Maxexact||𝑢()𝐴𝑝.(32)

Taking the logarithm of both sides of (32), we obtain𝐸log=log(𝐴)+𝑝log().(33)

For two different refined mesh spacing 1 and 2, we have the following two relations𝐸log1=log(𝐴)+𝑝log1,(34a)𝐸log2=log(𝐴)+𝑝log2.(34b)

Subtracting (34b) from (34a), we obtain the order (rate) of convergence 𝐸𝑝=log1𝐸log2log1log2,(35) where 𝐸1 and 𝐸2 are maximum absolute errors for two uniform mesh widths 1 and 2, respectively. For computation of order of convergence of the proposed method, we have considered 1=1/40 and 2=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+2) 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+4) accuracy for the solution of wave equation in polar coordinates. For a fixed parameter 𝛾=𝑘/2, 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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