A two-step explicit P-stable method for solving second order initial value problems

doi:10.1016/S0096-3003(02)00154-6

Applied Mathematics and Computation

Volume 138, Issues 2–3, 20 June 2003, Pages 435-442

https://doi.org/10.1016/S0096-3003(02)00154-6 Get rights and content

Abstract

The conceptions of periodicity interval and P-stability were introduced by Lambert and Watson [J. Inst. Math. Appl. 18 (1976) 189] in connection with multistep methods for second order initial value problems. This paper presents a new two-step explicit P-stable method of order two for solving initial value problems of second order ordinary differential equations. Based on a special vector operation, the method can be extended to be vector applicable.

Introduction

Let us consider the initial value problem of the second order ordinary differential equations $y″=f(x,y), y(x_{0})=y_{0}, y′(x_{0})=y′,$ where presume f(x,y) is sufficiently differentiable and the first derivative does not appear explicitly in f(x,y). The numerical methods have been paid much attention to in recent years because the problems are usually encountered in celestial mechanics, quantum mechanical scattering theory, theoretical physics and chemistry, and electronics. Generally, the solution of (1) is periodic, so it is expected that the result produced by some numerical method be of the analogical periodicity of the analytic solution. In 1976, Lambert and Watson [1] proposed the conceptions of periodicity interval and P-stability which can be used to discuss the stability of the numerical method for second order initial value problems. Although many P-stable methods have been proposed, such as linear multistep methods, high order hybrid P-stable methods, implicit Runge–Kutta–Nystrom and so on [2], [3], [4], [5], [6], these methods are implicit so there is an iteration subprocess needed in each step. Recently, some authors presented a class of one-step explicit A-stable methods of high order for stiff problems (see [7], [8], [9]), and with the aid of a vector operation, these methods can be extended to be vector applicable [10]. Motivated by the idea, we will present a new two-step explicit method of order two for solving (1). Based on the aforementioned special vector operation, our method can be extended to be vector applicable directly.

Section snippets

Preliminaries

To investigate the stability properties of methods for solving the initial value problem (1), Lambert and Watson [1] introduced the scalar test equation $y″=−ω^{2} y, ω∈R,$ When applying a symmetric two-step method to the test equation (2), one obtains the following difference equation of the form: $y_{n+1} −2C(H)y_{n} +y_{n−1} =0,$ where H=ωh,h is a fixed step length, C(H) is a rational polynomial with respect to H. The characteristic equation and polynomial are defined by the following respectively: $ξ^{2} −2C(H)ξ+1=0,$

New two-step explicit P-stable method

First of all, our analysis is restricted in one-dimension space. Consider the following formula for solving the initial value problem (1): $y_{n+1} +y_{n−1} =2y_{n} exp h^{2} f_{n} 2y_{n},$ where h is the step length and presume y_n≠0.

Applying (8) to the scalar test equation (2), one gets its characteristic equation $ξ^{2} −2 e^{−(H^{2}/2)} ξ+1=0,$ where H²=(ωh)². Obviously for all H²∈(0,+∞), e^−(H²/2)<1, thus method (8), as a symmetric method, is P-stable.

Let y(x) be the true solution of (1) and presume y_n, y_n−1 are precise. Consider the

The vector form of the method

Definition 4.1

[8]. Let a,b,d∈C^m, d≠0, the vector product and quotient, a special vector trinary operation, of a,b,d is defined by the following: $a∗ b d =b∗ a d := b Re (a,d)+a Re (b,d)−d Re (b,a) (d,d),$ where (·,·) is the inner product in C^m,Re(·) is the real part of complex number. Define the power of the operation by $d∗ a d^{n} := ⋯ n d∗ a d ∗ a d ∗⋯∗ a d$ and define the exponent of the operation by $d∗ e^{(a/d)} :=∑^{+∞}_{k=0} 1 k! d∗ a d^{k} .$ Obviously, , is well-defined. And if a,b,c,d∈C^m, d≠0, then $(k_{1} a+k_{2} b)∗ c k_{3} d = k_{1} k_{3} a∗ c d + k_{2} k_{3} b∗ c d, k_{1},k_{2},k_{3} ∈R, k_{3} ≠0$ and $d∗ a d =a, d∗ a d$

Numerical results

Problem 1

Consider the scalar test equation y″=−ω²y, y(0)=1, y′(0)=0. The true solution is y(x)=cos(ωx). Set ω=10. In the numerical experiment, we take the step length h=π/200,π/400,π/800,π/1600, and for simplicity, the true value at x=h is taken as the second started value. The following table gives the errors at point x=5π, 10π, 15π, 20π.

Point	$h= π 200$	$h= π 400$	$h= π 800$	$h= π 1600$
$x=5π$	$−$ 5.1541e $−$ 002	$−$ 3.2539e $−$ 003	$−$ 2.0362e $−$ 004	$−$ 1.2731e $−$ 005
$x=10π$	$−$ 2.0104e $−$ 001	$−$ 1.3001e $−$ 002	$−$ 8.1460e $−$ 004	$−$ 5.0928e $−$ 005
$x=15π$	$−$ 4.3307e $−$ 001	$−$ 2.9177e $−$ 002	$−$

Remark

The numerical results illustrate our method is efficient. Since it is explicit, the computational effort at each step is small relatively. However, the order of the method is low, so it needs to go on discussing how to improve its accuracy so as to adapt the big step length.

References (11)

M. Chawla et al.
Two-step fourth-order P-stable methods with phase lag of order six for y″=f(t,y)
J. Comput. Appl. Math.
(1986)
T.E. Simos
A four-step method for the numerical solution of the Schrodinger equation
J. Comput. Appl. Math.
(1990)
X.Y. Wu
A sixth-order A-stable explicit one-step method for stiff systems
Comput. Math. Appl.
(1998)
X.Y. Wu et al.
The vector form of a sixth-order A-stable explicit one-step method for stiff problems
Comput. Math. Appl.
(2000)
X.Y. Wu et al.
An explicit two-step method exact for the scalar test equation y′=λy
Comput. Math. Appl.
(2000)

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A two-step explicit P-stable method for solving second order initial value problems

Abstract

Introduction

Section snippets

Preliminaries

New two-step explicit P-stable method

The vector form of the method

Numerical results

Remark

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Comput. Math. Appl.

Comput. Math. Appl.

Comput. Math. Appl.