New L-stable method for numerical solutions of ordinary differential equations
Introduction
The initial-value problem (IVP) is known asand the general one-step multistep method for a numerical solution of (1.1) is given bywhere 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, and 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 bywithJacques [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]:For the first step, n is set to zero and the nonlinear equations (1.5), (1.6), (1.7) are solved using the iterationThe process is started by setting n = 0 in (1.8), (1.9), (1.10). An initial guess is made for This value together with initial conditions y0 is substituted into (1.8) to calculate Afterwards, this value together with y0 is substituted in (1.9) and is calculated. is now substituted into (1.10) to obtain an improved approximation This value is then substituted into (1.8), (1.9) to obtain and 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 obtainwhere ∣T1∣ ⩽ M1hk+1 for some positive constant M1. If Eq. (1.6) has an LTE of order k thenwhere ∣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,For this equation, (1.5), (1.6), (1.7) can be rewritten asThe elimination of and then giveswhere μ = λh and λ is a complex constant. R(μ) is the amplification factor which is given by
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
References (9)
- et al.
A-stable extended Trapezoidal rule for integration of ODEs
Comput. Math. Appl.
(1985) - et al.
A class of stabilised extended one-step methods for the numerical solution of ODEs
Comput. Math. Appl.
(1995) - et al.
Extended double-stride L-stable methods for the numerical solution of ODEs
Comput. Math. Appl.
(1996) - et al.
Extended A-stable two-step methods for the numerical solution of ODEs
Int. J. Comput. Math.
(1992)