New L-stable method for numerical solutions of ordinary differential equations

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Abstract

Recently, Kondrat and Jacques [D.M. Kondrat, I.B. Jacques, Extended A-stable two-step methods for the numerical solution of ODEs, Int. J. Comput. Math. 42 (1992) 117–124], Bulut and Inc [H. Bulut, M. Inc, A two-step method for the numerical integration of stiff differential equations, Int. J. Comput. Math. 73 (2000) 333–340] have reported on an extended two-step fourth-order A-stable method. In this paper we describe a new class of stabilised fourth-order L-stable method. Numerical results are presented for three test problems.

Introduction

The initial-value problem (IVP) is known asdydx=f(x,y);y(a)=A,axb,and the general one-step multistep method for a numerical solution of (1.1) is given byα0yn+α1yn+1=hβ0fn+β1fn+1,where αi, βi are fixed constants and fn = f(xn, yn), i = 0, 1.

It is well-known that yn is an approximation to y(xn) at a sequence of equally spaced points where xn = a + nh, h = step  length, n = 0, 1, 2, … The highest order A-stable method of this type is obtained by setting α0 = −1, α1 = 1, β0=12 and β1=12 in (1.2). This is the trapezoidal method and is second-order convergent. According to a well-known theorem suggested by Dahlquist [3], the order of a A-stable linear multistep method (LMM) cannot exceed two, and the optimal second-order A-stable LMM is the trapezoidal rule. Usmani and Agarwal [4] constructed a new one-step method by coupling together linear multistep formulas. This A-stable method is convergent of third-order and is defined byyˆn+1=5yn-4yn+1+h2fn+4fn+1withyn+1=yn+h125fn+8fn+1-fˆn+2.Jacques [5] modified the method of Usmani and Agarwal [4] to obtain a one-step third-order A-stable method which is also L-stable. Kondrat and Jacques [1] also presented extended two-step fourth-order A-stable methods. Chawla et al. [6] described fourth-order extended one-step methods which are A-stable or L-stable. In [7], they also introduced a class of extended one-step methods generalizing the method of Usmani and Agarwal [4]. Afterwards, Chawla et al. [8] showed that an extended double-stride fourth-order method for every interval is L-stable.

The purpose of this paper is to investigate the accuracy and stability of the method which is of the general form [8]:y˜n+1=14(1+4δ0)yn+14(3-4δ0)yn+2+hδ0fn+12(2δ0-1)fn+2,yˆn+1=α0yn+(1-α0)yn+2+h12(1+4α0)fn+8(2α0-1)f˜n+1+(4α0-5)fn+2,yn+2=yn+h3(β0fn+4β1fˆn+1+β2fn+2).For the first step, n is set to zero and the nonlinear equations (1.5), (1.6), (1.7) are solved using the iterationy˜n+1[m]=14yn+3yn+2[m]-h2fn+2[m],yˆn+1[m]=14yn+3yn+2[m]+h6fn-2f˜n+1[m]-2fn+2[m],yn+2[m+1]=yn+h3fn+4fˆn+1[m]+fn+2[m].The process is started by setting n = 0 in (1.8), (1.9), (1.10). An initial guess is made for y2[0]. This value together with initial conditions y0 is substituted into (1.8) to calculate y˜1[0]. Afterwards, this value together with y0 is substituted in (1.9) and yˆ1[0] is calculated. yˆ1[0] is now substituted into (1.10) to obtain an improved approximation y2[1]. This value is then substituted into (1.8), (1.9) to obtain y˜1[1] and yˆ1[1] and so on. Kondrat and Jacques [1] proved that the overall method was fourth-order convergent and A-stable. Therefore, their new method has a higher order of convergence than the method of Usmani and Agarwal [4].

In particular, in Sections 2 Order of convergence, 3 Extended fourth-order L-stable method, we indicate that the coefficients δ0, α0 and β0 can be chosen to construct a new method which is fourth-order convergent and L-stable. Finally, in Section 4 this method is used to solve ordinary differential equations. These results are compared with those of obtained by Kondrat and Jacques [1], the Usmani and Agarwal [4], Jacques [5] and the Runge–Kutta fourth-order methods.

Section snippets

Order of convergence

Suppose that LMM Eq. (1.7) has a local truncation error (LTE) of order (k + 1). If this is denoted by T1 and the exact solution of the initial-value problem (1.1) at Xn is denoted by Yn then we obtainYn+2=Yn+h3[f(Xn,Yn)+4f(Xn+1,Y^n+1)+f(Xn+2,Yn+2)]+T1,where ∣T1  M1hk+1 for some positive constant M1. If Eq. (1.6) has an LTE of order k thenY^n+1=α0Yn+(1-α0)Yn+2+h12[(1+4α0)f(Xn,Yn)+8(2α0-1)f˜(Xn+1,Yn+1)+(4α0-5)f(Xn+2,Yn+2)+T2,where ∣T2  M2hk for some positive constant M2.

Similarly, if Eq. (1.5) has

Extended fourth-order L-stable method

In order to examine the present method for the stability, let us consider the differential equation,y=λy,λ>0.For this equation, (1.5), (1.6), (1.7) can be rewritten asy˜n+1=14(1+4δ0)yn+14(3-4δ0)yn+2+λhδ0yn+12(2δ0-1)yn+2,yˆn+1=α0yn+(1-α0)yn+2+λh12[(1+4α0)yn+8(2α0-1)y˜n+1+(4α0-5)yn+2],yn+2=Yn+λh3[β0yn+4β1yˆn+1+β2yn+2].The elimination of y˜n+1 and yˆn+1 then givesyn+2=R(μ)yn,where μ = λh and λ is a complex constant. R(μ) is the amplification factor which is given byR(μ)=1+μ3(1+4δ0)+μ29[(1+4α0)+2(2α0

Numerical results

In this section, some numerical results are presented for three test problems and also their analytic solutions are calculated. The new method is compared with the Kondrat and Jacques [1], the Usmani and Agarwal [4], Jacques [5] and the Runge–Kutta fourth-order methods.

Our present method is based on Simpson’s rule and, those noted earlier. Simpson’s rule is an optimal fourth-order method among all linear two-step methods. Our present method is a class of extended methods expected to perform

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