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The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last.

Neutral delay differential equations (NDDEs) are widely used in various kinds of applied disciplines such as biology, ecology, electrodynamics, and physics and hence intrigue lots of researchers in numerical simultation and analysis (see, e.g., [

The present paper was in part inspired by the work of Spijker et al. With stepsize restriction to some numerical schemes, they revealed to us some monotonicity and stability properties for ODEs, respectively (see, [

The rest of the paper is organized as follows. In Section

In the present paper, we consider the following nonlinear NDDEs:

When

Now, let us consider

According to Liu in [

Now, let

A numerical method for DDEs or NDDEs is called globally stable, if there exists a constant

A numerical method for DDEs or NDDEs is said to be asymptotically stable, if

In the section, we will discuss the relationship between the stepsize restriction and nonlinear stability of the method.

Assume condition (

Let

Then, together with (

This implies that

Note that

Assume condition (

A Runge-Kutta method is algebraically stable if the matrix

Assume condition (

Assume condition (

Like in the proof of Theorem

Assume condition (

Assume condition (

As it is shown in the theorems, the parameters

Consider the following case, like the conditions in [

In particular, let

Next, we give some examples on how to calculate the parameter

Consider

Consider 2-stage 2-order Runge-Kutta method:

For more Runge-Kutta methods with the nonnegative definite matrix

In this study, we show that the Runge-Kutta methods with stepsize restrictions can preserve global and asymptotical stability of the continuous DDEs or NDDEs. An algebraically stable Runge-Kutta method with linear interpolation procedure for DDEs or NDDEs is globally stable and asymptotically stable. These results can be easily extended to the following equation with several delays:

Moreover, the present results have certain instructive effect in numerical simulation. In the future, we hope to apply the results to some real-world problems, for example, reaction-diffusion dynamical systems with time delay [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by NSFC (Grant nos. 11201161, 11171125, and 91130003).