Asymptotical stability of Runge–Kutta methods for nonlinear impulsive differential equations

In this paper, asymptotical stability of the exact solutions of nonlinear impulsive ordinary differential equations is studied under Lipschitz conditions. Under these conditions, asymptotical stability of Runge–Kutta methods is studied by the theory of Padé approximation. And two simple examples are given to illustrate the conclusions.


Introduction
The impulsive differential equations (IDEs) are widely applied in numerous fields of science and technology: theoretical physics, mechanics, population dynamics, pharmacokinetics, industrial robotics, chemical technology, biotechnology, economics, etc. Recently, the theory of IDEs has been an object of active research. Especially, stability of the exact solutions of IDEs has been widely studied (see [1,2,9,16,18] and the references therein). However, many IDEs cannot be solved analytically or their solving is more complicated. Hence taking numerical methods is a good choice.
In recent years, the stability of numerical methods for IDEs has attracted more and more attention (see [11,12,15,17,22,29] etc.). Stability of Runge-Kutta methods with the constant stepsize for scalar linear IDEs has been studied by [17]. Runge-Kutta methods with variable stepsizes for multidimensional linear IDEs has been investigated in [12]. Collocation methods for linear nonautonomous IDEs has been considered in [29]. An improved linear multistep method for linear IDEs has been investigated in [13]. Stability of the exact and numerical solutions of nonlinear IDEs has been studied by the Lyapunov method in [11]. Stability of Runge-Kutta methods for a special kind of nonlinear IDEs has been investigated by the properties of the differential equations without impulsive perturbations in [15]. Stability and asymptotic stability of implicit Euler method for stiff IDEs in Banach space has been studied by [22]. There is a lot of significant work on the numerical solution of impulsive differential equations, for example [6,7,10,14,[23][24][25][26][27]. However, in this work the authors did not investigate the stability of the numerical methods for non-stiff nonlinear IDEs under Lipschitz conditions. Consider the equation of the form where x(t + ) is the right limit of x(t), t 0 = τ 0 < τ 1 < τ 2 < · · · , lim k→∞ τ k = ∞, the function f : [t 0 , +∞) × C d → C d is continuous in t and Lipschitz continuous with respect to the second variable in the following sense: there is a positive real constant α such that where · is any convenient norm on C d . And also assume that each function I k , k = 1, 2, . . . is Lipschitz continuous i.e. there is a positive constant β k such that

Asymptotical stability of the exact solution
In this section, we study the asymptotical stability of the exact solution of (1). In order to investigate the asymptotical stability of x(t), consider Eq. (1) with another initial data: where Z + = {1, 2, . . .}.
The exact solution x(t) of (1) is said to be 1 stable if, for an arbitrary > 0, there exists a positive number δ = δ( ) such that, for any other solution y(t) of (4), x 0y 0 < δ implies 2 asymptotically stable, if it is stable and lim t→∞ x(t)y(t) = 0.

Theorem 2.2 Assume that there exists a positive constant γ such that
The exact solution of (1) is asymptotically stable if there is a positive constant C such that for arbitrary k ∈ Z + .
From the proof of Theorem 2.2, we can obtain the following result.

Remark 2.3 If the condition (5) of Theorem 2.2 is changed into the weaker condition
then the exact solution of (1) is stable.

Runge-Kutta methods
In this section, Runge-Kutta methods for (1) can be constructed as follows: where h k = τ k+1 -τ k m , t k,l = τ k + lh k , t i k,l = t k,l + c i h k , x k,l is an approximation to the exact solution x(t k,l ) and X i k,l is an approximation to the exact solution . . , s, s is referred to as the number of stages. The weights b i , the abscissae c i = s j=1 a ij and the matrix A = [a ij ] s i,j=1 will be denoted by (A, b, c). Similarly, the Runge-Kutta methods for (4) can be constructed as follows:  (8), x k,ly k,l ; 2 asymptotically stable, if it is stable and if ∃M 1 > 0, for any m ≥ M 1 , h k = τ k+1 -τ k m , k ∈ N, the following holds: where with error It is the unique rational approximation to e z of order j + k, such that the degrees of numerator and denominator are j and k, respectively.

Theorem 3.8 Under the conditions of Theorem
Proof Obviously, we can obtain which implies Therefore, by Lemma 3.7 and the method of introduction, we obtain

Numerical experiments
In this section, two simple numerical examples in real space are given.
Therefore, by Theorem 2.2, the exact solution of (12) is asymptotically stable. By Corollary 3.6, the explicit Euler method (see Fig. 3) and classical 4-stage fourth order explicit Runge-Kutta methods (see Fig. 4) for (12) are asymptotically stable for h k = 1 m , k ∈ N, m is an arbitrary positive integer.
From Tables 1 and 2, we can see that the Runge-Kutta methods conserve their orders of convergence.