Two Schemes of Impulsive Runge–Kutta Methods for Linear Differential Equations with Delayed Impulses

Abstract

In this paper, two different schemes of impulsive Runge–Kutta methods are constructed for a class of linear differential equations with delayed impulses. One scheme is convergent of order p if the corresponding Runge–Kutta method is p order. Another one in the general case is only convergent of order 1, but it is more concise and may suit for more complex differential equations with delayed impulses. Moreover, asymptotical stability conditions for the exact solution and numerical solutions are obtained, respectively. Finally, some numerical examples are provided to confirm the theoretical results.

1. Introduction

Impulsive differential equations (IDEs) arise widely in numerous mathematics models of systems with instantaneous perturbations. Such models have been applied with huge success in lots of application fields, such as control theory, medicine, biotechnology, economics, population growth, etc. Some work on these systems is presented in [1,2,3,4]. Recently, more and more experts and scholars have begun to pay attention to different kinds of differential equations with delayed impulses (DEDIs) and have achieved many important results about the following equations: nonlinear ordinary differential equations with delayed impulses (see [4,5,6,7,8], etc.), time-delay differential equations with delayed impulses [4,9,10,11,12,13,14], and stochastic differential equations with delayed impulses [15,16,17,18,19,20]. In the present paper, asymptotical stability conditions for the zero solution of a class of linear DEDIs are obtained, and two interesting examples are provided to reveal the effect of delayed impulses on differential equations, which can potentially destabilize a stable system or stabilize an unstable one.

Recently, the theory of numerical methods for impulsive differential equations has also been developing rapidly. The convergence and stability of impulsive Runge–Kutta methods for scalar linear IDEs [21,22], multidimensional linear IDEs [23], semilinear IDEs [24], nonlinear IDEs [25,26,27,28,29], impulsive time-delay differential equations [30,31,32,33,34,35], and stochastic impulsive time-delay differential equations [36] have been studied, respectively. There is a lot of important and relevant literature (see [37,38,39,40,41,42], etc.). However, to our knowledge, most of the previous literature focused on numerical methods for IDEs or impulsive time-delay differential equations; the research on numerical methods for DEDIs is still lacking.

The rest of this paper is organized as follows. In Section 2, asymptotical stability conditions for the zero solution of a class of linear DEDIs and two examples are given to show how delayed impulsive actions influence the stability of the zero solution of the equations. In Section 3, the scheme 1 impulsive Runge–Kutta methods (S1IRKMs) are constructed, and the convergence and asymptotical stability of the methods are studied. Moreover, the asymptotical stability of the scheme 1 impulsive $\theta$ method (S1I$\theta$M) is studied. In Section 4, the scheme 2 impulsive Runge–Kutta methods (S2IRKMs) are constructed based on classical Runge–Kutta methods, and the convergence and stability of the methods are studied. In general, a S2IRKM is only convergent of order 1. Therefore, it is very necessary to consider the scheme 2 impulsive $\theta$ method (S2I$\theta$M). Moreover, the asymptotical stability of S2I$\theta$Ms is studied. In Section 5, we provide some numerical examples to confirm our theoretical results. Finally, in Section 6, conclusions and future work are provided.

2. Asymptotical Stability of DEDIs

In this section, not only are the asymptotical stability conditions for the zero solution of DEDIs obtained but also two examples are given to illustrate that delayed impulses can change a previously unstable problem into a stable one or a previously stable problem into an unstable one.

2.1. Asymptotical Stability of DEDIs

In this paper, we consider the impulsive differential equation:

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=ax\left(t\right),\hfill & t\ge 0,t\ne k\tau ,k\in {\mathbb{Z}}^{+}=\{1,2,\cdots \},\hfill \\ x\left(k{\tau }^{+}\right)=bx\left(r_k\right),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0,\hfill \end{array}\right.$$

(1)

where $a\ne 0$, b, $x_0$, $\tau$, and $r_k$ are real constants, $b\ne 0$, $k\in {\mathbb{Z}}^{+}$. There is a real constant $\sigma \in (0,1]$, such that all $r_k,k\in {\mathbb{Z}}^{+}$ satisfy

$$r_k=\sigma k\tau +(1-\sigma )(k-1)\tau .$$

Definition 1. 

$x\left(t\right)$ is said to be the solution of (1) if

1. 

${lim}_{t\to 0^{+}}x\left(t\right)=x(0+0)=x_0$,

2. 

For $t\in (0,+\infty )$, $t\ne k\tau$, $x\left(t\right)$ is differentiable and satisfies $x^{\prime }\left(t\right)=ax\left(t\right)$,

3. 

$x\left(t\right)$ is left-continuous in $(0,+\infty )$ and if $t=k\tau$, then $x\left(k{\tau }^{+}\right)=bx\left(r_k\right)$, where $x\left(k{\tau }^{+}\right)={lim}_{t\to k{\tau }^{+}}x\left(t\right)$.

Problem (1) has a unique solution as follows:

$$x\left(t\right)={\left(b{\mathrm{e}}^{a\sigma \tau }\right)}^k{x}_0{\mathrm{e}}^{a(t-k\tau )},\forall t\in (k\tau ,(k+1)\tau ].$$

(2)

From (2), it is easy to obtain the following theorem.

Theorem 1.

The solution $x\left(t\right)\equiv 0$ of (1) is asymptotically stable ($x\left(t\right)\to 0$ as $t\to +\infty$) if and only if

$$\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1.$$

(3)

When $\sigma =1$, DEDI (1) is changed into an IDE (not delayed). Consequently, when $\sigma =1$, the necessary and sufficient condition (3) for the asymptotical stability of DEDI (1) (Theorem 1) is changed into the special case of the necessary and sufficient condition for the asymptotical stability of an IDE (not delayed). Hence the result of ([21] Theorem 1.4) is the special case of Theorem 1 of the present paper. The difference between the asymptotical stability of the DEDI and the asymptotical stability of the IDE also can be seen from the following two examples.

2.2. Two Interesting Examples

In this subsection, we present two differential equations with delayed impulses which are interesting and offer simple examples to show how the delayed impulsive actions influence the stability of the zero solution of the equations. In Example 1, the zero solutions of an ordinary differential equation and an impulsive differential equation are unstable, but the same equation with delayed impulses is asymptotically stable. Conversely, in Example 2, the zero solutions of an ordinary differential equation and an impulsive differential equation are asymptotically stable, but the same equation with delayed impulses is unstable.

Example 1.

First, consider the following simple scalar ordinary differential equation:

$$x^{\prime }\left(t\right)=x\left(t\right),t\ge 0,x\left(0\right)=x_0.$$

(4)

Solving this equation, we can obtain the exact solution of (4),

$$x\left(t\right)=x_0{\mathrm{e}}^t,t\ge 0,$$

which implies that when $x_0\ne 0$,

$$\underset{t\to +\infty }{lim}x\left(t\right)=+\infty ,$$

which also implies that the zero solution of (4) is unstable.

Second, consider the same equation with impulses (not delayed):

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=x\left(t\right),\hfill & t\ge 0,t\ne k,k\in {\mathbb{Z}}^{+},\hfill \\ x\left(k^{+}\right)=\left(\frac{1}{2}\right)x\left(k\right),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0.\hfill \end{array}\right.$$

(5)

Solving this equation, we can obtain the exact solution of (5),

$$x\left(t\right)={\left(0.5\mathrm{e}\right)}^k{x}_0{\mathrm{e}}^{t-k},\forall t\in (k,k+1].$$

which implies that when $x_0\ne 0$,

$$\underset{t\to +\infty }{lim}x\left(t\right)=+\infty ,$$

which also implies that the zero solution of (5) is unstable.

Finally, consider the same differential equation with delayed impulses:

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=x\left(t\right),\hfill & t\ge 0,t\ne k,k\in {\mathbb{Z}}^{+},\hfill \\ x\left(k^{+}\right)=\left(\frac{1}{2}\right)x(k-\frac{9}{10}),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0.\hfill \end{array}\right.$$

(6)

Solving this equation, we can obtain the exact solution of (6),

$$x\left(t\right)={\left(0.5{\mathrm{e}}^{\frac{1}{10}}\right)}^k{x}_0{\mathrm{e}}^{t-k},\forall t\in (k,k+1].$$

By Theorem 1 of the present paper, we can obtain that the zero solution of (6) is asymptotically stable (see Figure 1).

Example 2.

First, consider the following simple scalar ordinary differential equation:

$$x^{\prime }\left(t\right)=-x\left(t\right),t\ge 0,x\left(0\right)=x_0.$$

(7)

Solving this equation, we can obtain the exact solution of (7),

$$x\left(t\right)=x_0{\mathrm{e}}^{-t},t\ge 0.$$

Obviously, the zero solution of (7) is asymptotically stable.

Second, consider the same differential equation with impulses (not delayed):

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=-x\left(t\right),\hfill & t\ge 0,t\ne k,k\in {\mathbb{Z}}^{+},\hfill \\ x\left(k^{+}\right)=2x\left(k\right),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0.\hfill \end{array}\right.$$

(8)

Solving this equation, we can obtain the exact solution of (5),

$$x\left(t\right)={\left(\frac{2}{\mathrm{e}}\right)}^k{x}_0{\mathrm{e}}^{t-k},\forall t\in (k,k+1].$$

By ([21] Theorem 1.4), we can obtain that the zero solution of (5) is asymptotically stable.

Finally, consider the following differential equation with delayed impulses:

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=-x\left(t\right),\hfill & t\ge 0,t\ne k,k\in {\mathbb{Z}}^{+},\hfill \\ x\left(k^{+}\right)=2x(k-\frac{9}{10}),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0.\hfill \end{array}\right.$$

(9)

Solving this equation, we can obtain the exact solution of (6),

$$x\left(t\right)={\left(2{\mathrm{e}}^{-\frac{1}{10}}\right)}^k{x}_0{\mathrm{e}}^{t-k},\forall t\in (k,k+1].$$

which implies that when $x_0\ne 0$,

$$\underset{t\to +\infty }{lim}x\left(t\right)=+\infty ,$$

which implies that the zero solution of (9) is unstable (see Figure 2).

3. S1IRKM for (1)

The special case of $\sigma =1$ has already been studied in paper [21], and below we focus on the case of $0<\sigma <1$. All the points in the set $S=\{k\tau ,r_k:k\in {\mathbb{Z}}^{+}\}$ are chosen as the numerical mesh. For convenience, we divide the intervals $[(k-1)\tau ,r_k]$ and $[r_k,k\tau ]$ ($k\in {\mathbb{Z}}^{+}$) equally by m, where m is a positive integer, respectively. That means the step sizes are as follows for $\forall k\in \mathbb{N}=\{0,1,2,\cdots \}$:

$$h_{k,l}=\left\{\begin{array}{cc}{\overline{h}}_1:=\frac{r_k-(k-1)\tau }{m}=\frac{\sigma \tau }{m},\hfill & l=1,2,\cdots ,m,\hfill \\ {\overline{h}}_2:=\frac{k\tau -r_k}{m}=\frac{(1-\sigma )\tau }{m},\hfill & l=m+1,m+2,\cdots ,2m.\hfill \end{array}\right.$$

The mesh point $t_{k,0}=k\tau$, $t_{k,l}=k\tau +{\displaystyle \sum _{j=0}^l}{h}_{k,j}$, $\forall k\in \mathbb{N}$, $l=1,2,\cdots ,2m$.

$$\left\{\begin{array}{cc}{x}_{k,l+1}=x_{k,l}+a{h}_{k,l+1}{\displaystyle \sum _{i=1}^v}{b}_i{X}_{k,l+1}^i,\hfill & l=0,1,\cdots ,2m-1,\hfill \\ X_{k,l+1}^i=x_{k,l}+a{h}_{k,l+1}{\displaystyle \sum _{j=1}^v}{a}_{ij}{X}_{k,l+1}^j,\hfill & i=1,2,\cdots ,v,\hfill \\ x_{k+1,0}=b{x}_{k,m},\hfill & k\in \mathbb{N},\hfill \\ x_{0,0}=x_0,\hfill \end{array}\right.$$

(10)

where v refers to the number of stages. The weights $b_i$, the abscissae $c_i={\sum }_{j=1}^v{a}_{ij}$, and the matrix $A={\left[a_{ij}\right]}_{j=1}^v$ are denoted by $(A,b,c)$. We denote the approximation to the solution $x\left(t_{k,l}\right)$, $x\left(r_k\right)$, and $x(k\tau +0)$ by $x_{k,l}$ ($l=1,2,\cdots ,2m$), $x_{k,m}$, and $x_{k,0}$, respectively.

Equation (10) can be written as

$$\left\{\begin{array}{cc}{x}_{k,l+1}=\left(1+z_1{b}^T{(I-z_{1}A)}^{-1}e\right)x_{k,l}=R\left(z_1\right)x_{k,l},\hfill & l=0,1,\cdots ,m-1,\hfill \\ x_{k,l+1}=\left(1+z_2{b}^T{(I-z_{2}A)}^{-1}e\right)x_{k,l}=R\left(z_2\right)x_{k,l},\hfill & l=m,\cdots ,2m-1,\hfill \\ x_{k+1,0}=b{x}_{k,m},\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x_{0,0}=x_0,\hfill \end{array}\right.$$

(11)

where $R\left(z\right)=1+z{b}^T{(I-zA)}^{-1}e$, $z_1=a{\overline{h}}_1=\frac{a\sigma \tau }{m}$ and $z_2=a{\overline{h}}_2=\frac{a(1-\sigma )\tau }{m}$.

3.1. Asymptotical Stability of S1IRKMs

Theorem 2. 

Assume the condition (3) holds, and the stability function of the Runge–Kutta method is $R\left(z\right)=\frac{Q_r\left(z\right)}{P_s\left(z\right)}$, which is given by the $(r,s)$-Padé approximation to ${\mathrm{e}}^z$, $\left|z\right|<1$ for $z=z_1$, and $z=z_2$:

(i) 

if $a>0$ and s is even, then S1IRKM (10) for (1) is asymptotically stable,

(ii) 

if $a<0$ and r is odd, then S1IRKM (10) for (1) is asymptotically stable.

Proof. 

From scheme (11), we can obtain that for $\forall k\in \mathbb{N}$, $0\le l\le m$,

$$x_{k,l}={\left(b{\left(R\left(z_1\right)\right)}^m\right)}^k{x}_0{\left(R\left(z_1\right)\right)}^l,$$

and for $\forall k\in \mathbb{N}$, $m+1\le l\le 2m$,

$$x_{k,l}={\left(b{\left(R\left(z_1\right)\right)}^m\right)}^k{x}_0{\left(R\left(z_1\right)\right)}^m{\left(R\left(z_2\right)\right)}^{l-m}.$$

Hence, the numerical method (11) is asymptotically stable if and only if

$$|b{\left(R\left(z_1\right)\right)}^m|<1.$$

(12)

(i) If $a>0$ and s is even, applying ([21] Lemmas 3.3 and 3.7) and the condition (3), we can obtain

$$\left|b\right|{\left(R\left(z_1\right)\right)}^m\le \left|b\right|{\mathrm{e}}^{z_{1}m}=\left|b\right|{\mathrm{e}}^{a{\overline{h}}_{1}m}=\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1,$$

which implies that (12) holds.

(ii) Similarly, if $a<0$ and r is odd, applying ([21] Lemmas 3.3 and 3.7) and the condition (3), we can obtain

$$\left|b\right|{\left(R\left(z_1\right)\right)}^m\le \left|b\right|{\mathrm{e}}^{z_{1}m}=\left|b\right|{\mathrm{e}}^{a{\overline{h}}_{1}m}=\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1,$$

which implies that (12) holds. □

3.2. Convergence of S1RKM

In order to study the convergence of an S1RKM, the case where DEDI (1) is defined in the interval $[0,T]$ is considered in this subsection. For convenience, assume that there exists a positive integer N such that $T=N\tau$.

Lemma 1

([21,43,44,45]). There exists a unique $(r,s)$-Padé approximation $R_{rs}\left(z\right)=\frac{Q_r\left(z\right)}{P_s\left(z\right)}$ to ${\mathrm{e}}^z$ for $(r,s)\in \mathbb{N}\times \mathbb{N}$. Furthermore,

$$e^z{P}_s\left(z\right)-Q_r\left(z\right)=\frac{{(-1)}^s{z}^{r+s+1}}{(r+s)!}{\int }_0^1{u}^s{(1-u)}^r{\mathrm{e}}^{uz}du,$$

where

$$Q_r\left(z\right)=\frac{r!}{(r+s)!}\sum _{j=0}^r\frac{(r+s-j)!}{j!(r-j)!}{z}^j,$$

$$P_s\left(z\right)=\frac{s!}{(r+s)!}\sum _{j=0}^s\frac{(r+s-j)!}{j!(s-j)!}{(-z)}^j.$$

In order to analyze the local truncation errors of S1IRKM (10) for DEDI (1), consider the following problem:

$$\left\{\begin{array}{cc}{z}_{k,l+1}=z_{k,l}+a{h}_{k,l+1}{\displaystyle \sum _{i=1}^v}{b}_i{Z}_{k,l+1}^i,\hfill & l=0,1,\cdots ,2m-1,\hfill \\ Z_{k,l+1}^i=z_{k,l}+a{h}_{k,l+1}{\displaystyle \sum _{j=1}^v}{a}_{ij}{Z}_{k,l+1}^j,\hfill & i=1,2,\cdots ,v,\hfill \end{array}\right.$$

(13)

where $z_{k,0}=x\left(k{\tau }^{+}\right)$, $z_{k,l}=x\left(t_{k,l}\right)$, $k=0,1,2,·,N$, $l=1,2,·,2m-1$.

Theorem 3.

If the corresponding Runge–Kutta method is convergent of order p, then the local truncation errors between (13) and DEDI (1) satisfy that there exists a constant C such that for arbitrary $k=0,1,2,·,N$, $l=1,2,·,2m-1$,

$$R_{k,l+1}:=|z_{k,l+1}-x\left(t_{k,l+1}\right)|\le C{h}_{k,l+1}^{p+1}.$$

Proof. 

Because Runge–Kutta methods are convergent of order p, by Lemma 1, there exists a constant $C_1>0$ such that

$$R_{k,l+1}:=|{\mathrm{e}}^{a{h}_{k,l+1}}-R\left(a{h}_{k,l+1}\right)|\le C_1{h}_{k,l+1}^{p+1}.$$

(14)

Obviously, (13) can be rewritten as

$$z_{k,l+1}=R\left(a{h}_{k,l+1}\right)z_{k,l},$$

where $R_{rs}\left(a{h}_{k,l+1}\right)=(1+a{h}_{k,l+1}{b}^T(I-a{h}_{k,l+1}Ae))$. From the expression (2) for the solution $x\left(t\right)$ of DEDI (1), we have

$$\underset{t\in (0,T]}{sup}\left|x\left(t\right)\right|\le M.$$

Hence, the local errors satisfy

$$R_{k,l}=|x\left(t_{k,l+1}\right)-z_{k,l+1}|\le |{\mathrm{e}}^{a{h}_{k,l+1}}-R\left(a{h}_{k,l+1}\right)\left|\right|z_{k,l}|\le C{h}_{k,l+1}^{p+1}.$$

where $C=C_{1}M$. □

Theorem 4.

If Runge–Kutta methods are convergent of order p, then S1IRKM (10) for (1) is also convergent of order p, and in the following sense, there exists a constant $C_5$ such that for all $k\in \mathbb{N}$, $l=1,2,\cdots ,m$, the global errors satisfy

$$e_{k,l}=|x\left(t_{k,l}\right)-x_{k,l}|\le C_5{h}^p,$$

where $h=max\{{\overline{h}}_1,{\overline{h}}_2\}=\underset{k,l}{max}\left\{h_{k,l}\right\}$.

Proof. 

From (1) and (13), we have

$$\begin{array}{ccc}\hfill |X_{k,l+1}^i-Z_{k,l+1}^i|& \le & |x_{k,l}-z_{k,l}|+|a|h_{k,l+1}\left(\sum _{j=1}^v|a_{ij}\left|\right|X_{k,l+1}^i-Z_{k,l}^i|\right)\hfill \\ & \le & |x_{k,l}-z_{k,l}|+|a|h\left(\underset{1\le i\le v}{max}\sum _{j=1}^v\left|a_{ij}\right|\right)\underset{1\le i\le v}{max}\left\{\right|X_{k,l+1}^i-Z_{k,l}^i\left|\right\}\hfill \end{array}$$

which implies

$$\underset{1\le i\le v}{max}\left\{\right|X_{k,l+1}^i-Z_{k,l}^i\left|\right\}\le \Lambda |x_{k,l}-z_{k,l}|$$

where $\Lambda ={\left(1-\left|a\right|h\left({max}_{1\le i\le v}{\sum }_{j=1}^v\left|a_{ij}\right|\right)\right)}^{-1}$:

$$\begin{array}{ccc}\hfill |x_{k,l+1}-z_{k,l+1}|& \le & |x_{k,l}-z_{k,l}|+|a|h_{k,l+1}\left(\sum _{i=1}^v\left|b_i\right|\right)\underset{1\le i\le v}{max}\left\{\right|X_{k,l+1}^i-Z_{k,l}^i\left|\right\}\hfill \\ & \le & (1+\beta \Lambda |a|h_{k,l+1}\left)\right|x_{k,l}-z_{k,l}|\hfill \end{array}$$

where $\beta =\left({\displaystyle \sum _{i=1}^v}\left|b_i\right|\right)$. From Theorem 3, we have

$$R_1:=\underset{0\le k\le N,1\le l\le m}{max}\left\{R_{k,l}\right\}\le C{\overline{h}}_1{h}^p$$

and

$$R_2:=\underset{0\le k\le N,m+1\le l\le 2m}{max}\left\{R_{k,l}\right\}\le C{\overline{h}}_2{h}^p.$$

If $0\le l\le m-1$,

$$\begin{array}{cc}{e}_{k,l+1}\hfill & :=|x\left(t_{k,l+1}\right)-x_{k,l+1}|\hfill \\ & \le |x\left(t_{k,l+1}\right)-z_{k,l+1}|+|z_{k,l+1}-x_{k,l+1}|\hfill \\ \hfill & \le (1+\beta \Lambda |a|h_{k,l+1}\left)\right|x_{k,l}-z_{k,l}|+R_{k,l+1}\hfill \\ \hfill & \le (1+\beta \Lambda |a|{\overline{h}}_1)e_{k,l}+R_1\hfill \\ \hfill & \le {\left(1+\beta \Lambda \left|a\right|{\overline{h}}_1\right)}^{l+1}{e}_{k,0}+\left[{\left(1+\beta \Lambda \left|a\right|{\overline{h}}_1\right)}^{l+1}-1\right]\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \\ \hfill & \le {\mathrm{e}}^{(l+1)\beta \Lambda \left|a\right|{\overline{h}}_1}{e}_{k,0}+\left({\mathrm{e}}^{(l+1)\beta \Lambda \left|a\right|{\overline{h}}_1}-1\right)\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \\ \hfill & \le {\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }{e}_{k,0}+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \end{array}$$

(15)

or else, if $m\le l\le 2m-1$,

$$\begin{array}{cc}{e}_{k,l+1}\hfill & =|x\left(t_{k,l+1}\right)-x_{k,l+1}|\hfill \\ & \le |x\left(t_{k,l+1}\right)-z_{k,l+1}|+|z_{k,l+1}-x_{k,l+1}|\hfill \\ \hfill & \le (1+\beta \Lambda |a|h_{k,l+1}\left)\right|x_{k,l}-z_{k,l}|+R_{k,l+1}\hfill \\ \hfill & \le (1+\beta \Lambda |a|{\overline{h}}_2)e_{k,l}+R_2\hfill \\ \hfill & \le {\left(1+\beta \Lambda \left|a\right|{\overline{h}}_2\right)}^{l-m+1}{e}_{k,m}+\left[{\left(1+\beta \Lambda \left|a\right|{\overline{h}}_2\right)}^{l-m+1}-1\right]\frac{R_2}{\beta \Lambda \left|a\right|{\overline{h}}_2}\hfill \\ \hfill & \le {\mathrm{e}}^{(l-m+1)\beta \Lambda \left|a\right|{\overline{h}}_2}{e}_{k,m}+\left({\mathrm{e}}^{(l-m+1)\beta \Lambda \left|a\right|{\overline{h}}_2}-1\right)\frac{R_2}{\beta \Lambda \left|a\right|{\overline{h}}_2}\hfill \\ \hfill & \le {\mathrm{e}}^{\beta \Lambda \left|a\right|(1-\sigma )\tau }{e}_{k,0}+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|(1-\sigma )\tau }-1\right)\frac{R_2}{\beta \Lambda \left|a\right|{\overline{h}}_2}\hfill \end{array}$$

(16)

otherwise,

$$\begin{array}{cc}{e}_{k+1,0}\hfill & =|x\left(t_{k+1,0}\right)-x_{k+1,0}|\hfill \\ & =|bx\left(r_k\right)-b{x}_{k,m}|\le |b|e_{k,m}\hfill \\ \hfill & \le \left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|m{\overline{h}}_1}{e}_{k,0}+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|m{\overline{h}}_1}-1\right)\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \\ \hfill & =\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }{e}_{k,0}+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \\ \hfill & \le {\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}{e}_{0,0}+\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \end{array}$$

(17)

Because $e_{0,0}=0$, i.e., $x_{0,0}=x\left(0^{+}\right)=x_0$, it follows from (17) that we can obtain that for arbitrary $k=0,1,·,N-1$,

$$\begin{array}{cc}{e}_{k+1,0}\hfill & \le \frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R_1}{\beta \Lambda \left|a\right|{\overline{h}}_1}\hfill \\ \hfill & \le \frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{C{\overline{h}}_1^p}{\beta \Lambda \left|a\right|}\hfill \\ \hfill & \le C_2{h}^p,\hfill \end{array}$$

(18)

where $C_2=\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{C}{\beta \Lambda \left|a\right|}$. From (15) and (18), applying Theorem 3, we can obtain that for arbitrary $k=0,1,2,\cdots ,N-1$, $1\le l\le m$,

$$e_{k,l}\le C_3{h}^p,$$

(19)

where $C_3={\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }{C}_2+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{C}{\beta \Lambda \left|a\right|}$. Similarly, from (16) and (19), applying Theorem 3, we can obtain that for arbitrary $k=0,1,2,\cdots ,N-1$, $m\le l\le 2m$,

$$e_{k,l}\le C_4{h}^p,$$

(20)

where $C_4={\mathrm{e}}^{\beta \Lambda \left|a\right|(1-\sigma )\tau }{C}_3+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|(1-\sigma )\tau }-1\right)\frac{C}{\beta \Lambda \left|a\right|}$.

Finally, summarizing Equations (18)–(20), we know that all the global errors satisfy

$$e_{k,l}\le C_5{h}^p,\forall k=0,1,2,\cdots ,N-1,\forall l=0,1,2,\cdots ,2m,$$

where $C_5=max\{C_2,C_3,C_4\}$. □

3.3. Asymptotical Stability of S1I$\theta$Ms

Similarly, the scheme 1 impulsive $\theta$ method (S1I$\theta$M) for (1) can be constructed as follows:

$$\left\{\begin{array}{cc}{x}_{k,l+1}=x_{k,l}+h_{k,l}(a(1-\theta )x_{k,l}+a\theta x_{k,l+1}),\hfill & l=0,1,\cdots ,2m-1,\hfill \\ x_{k+1,0}=b{x}_{k,m},\hfill & k\in \mathbb{N},\hfill \\ x_{0,0}=x_0.\hfill \end{array}\right.$$

(21)

Obviously, S1I$\theta$M (21) can be written as

$$\left\{\begin{array}{cc}{x}_{k,l+1}=\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)x_{k,l}=\left(\frac{1+(1-\theta )z_1}{1-\theta z_1}\right)x_{k,l},\hfill & l=0,1,\cdots ,m-1,\hfill \\ x_{k,l+1}=\left(\frac{1+(1-\theta )a{\overline{h}}_2}{1-\theta a{\overline{h}}_2}\right)x_{k,l}=\left(\frac{1+(1-\theta )z_2}{1-\theta z_2}\right)x_{k,l},\hfill & l=m,\cdots ,2m-1,\hfill \\ x_{k+1,0}=b{x}_{k,m},\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x_{0,0}=x_0.\hfill \end{array}\right.$$

(22)

Theorem 5. 

Assume the condition (3) holds, $|z_1|<1$, and $|z_2|<1$:

(i) 

if $a>0$ and $0<\theta <\phi \left(1\right)$, then the impulsive θ method (21) for (1) is asymptotically stable,

(ii) 

if $a<0$ and $0<\theta <\phi \left(0\right)$, then the impulsive θ method (21) for (1) is asymptotically stable,

where $\phi \left(x\right)=\frac{1}{x}-\frac{1}{{\mathrm{e}}^x-1}$, $x\in \mathbb{R}$. (The function of φ can be referred to in Lemma 2 in ref. [46]).

Proof. 

From scheme (22), we can obtain that for $\forall k\in \mathbb{N}$, $0\le l\le m$,

$$x_{k,l}={\left(b{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^m\right)}^k{x}_0{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^l,$$

and for $\forall k\in \mathbb{N}$, $m+1\le l\le 2m$,

$$x_{k,l}={\left(b{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^m\right)}^k{x}_0{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^m{\left(\frac{1+(1-\theta )a{\overline{h}}_2}{1-\theta a{\overline{h}}_2}\right)}^{l-m}.$$

Hence, the numerical method (22) is asymptotically stable if and only if

$$|b{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^m|<1.$$

(23)

(i) If $a>0$ and $0<\theta <\phi \left(1\right)$, applying ([21] Lemma 2.3) and the condition (3), we can obtain

$$\left|b\right|{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^m\le \left|b\right|{\mathrm{e}}^{a{\overline{h}}_{1}m}=\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1,$$

which implies that (33) holds.

(ii) Similarly, if $a<0$ and $0<\theta <\phi \left(0\right)$, applying ([21] Lemma 2.3) and the condition (3), we can obtain

$$\left|b\right|{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^m\le \left|b\right|{\mathrm{e}}^{a{\overline{h}}_{1}m}=\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1,$$

which implies that (33) holds. □

4. S2IRKMs for (1)

For the second scheme, we pay attention to a uniform grid with step size $h=\frac{\tau }{m}$, where m is an integer. So, the formula for the time points is

$$t_{k,l}=k\tau +lh,k\in N,l=0,1,2,\cdots ,m.$$

The impulsive Runge–Kutta method for (1) can be constructed as follows:

$$\left\{\begin{array}{cc}{y}_{k,l+1}=y_{k,l}+ah{\displaystyle \sum _{i=1}^s}{b}_i{Y}_{k,l+1}^i,\hfill & l=0,1,\cdots ,m-1,\hfill \\ Y_{k,l+1}^i=y_{k,l}+ah{\displaystyle \sum _{j=1}^s}{a}_{ij}{Y}_{k,l+1}^j,\hfill & i=1,2,\cdots ,s,\hfill \\ y_{k+1,0}=b{y}_{k,\lfloor m\sigma \rfloor },\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ y_{0,0}=x_0.\hfill \end{array}\right.$$

(24)

Here, $y_{k,l}$ is an approximation of the exact solution $x\left(t_{k,l}\right)$, $\forall k\in \mathbb{N}$, $\forall l=1,2,\cdots ,m$. $y_{k,0}$ is an approximation of $x(k\tau +0)$, $\forall k\in {\mathbb{Z}}^{+}$. Obviously, if $\sigma m$ is an integer, $x_{k,\lfloor \sigma m\rfloor }$ is an approximation of the exact solution $x\left(r_k\right)$. Otherwise, we cannot find the numerical solutions at $t=r_k$. Now, $x_{k,\lfloor \sigma m\rfloor }$, which is an approximation of $x\left(t_{k,\lfloor \sigma m\rfloor }\right)$ ($t_{k,\lfloor \sigma m\rfloor }\le r_k$ and $|r_k-t_{k,\lfloor \sigma m\rfloor }|\le h$), is viewed as an approximation of $x\left(r_k\right)$ to find the numerical solution of (1).

The impulsive Runge–Kutta method (24) can be written as

$$\left\{\begin{array}{cc}{y}_{k,l+1}=\left(1+z{b}^T{(I-zA)}^{-1}e\right)y_{k,l}=R\left(z\right)y_{k,l},\hfill & l=0,1,\cdots ,m-1,\hfill \\ y_{k+1,0}=b{y}_{k,\lfloor m\sigma \rfloor },\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ y_{0,0}=x_0,\hfill \end{array}\right.$$

(25)

where $z=ha$.

4.1. Asymptotical Stability of Scheme 2 Impulsive Runge–Kutta Methods

Theorem 6. 

Assume the condition (3) holds, and the stability function of the Runge–Kutta method is $R\left(z\right)=\frac{Q_r\left(z\right)}{P_s\left(z\right)}$, which is given by the $(r,s)$-Padé approximation to ${\mathrm{e}}^z$, $z=ah$, $\left|z\right|<1$:

(i) 

if $a>0$ and s is even, then the impulsive Runge–Kutta method (24) for (1) is asymptotically stable,

(ii) 

if $a<0$ and r is odd, then the impulsive Runge–Kutta method (24) for (1) is asymptotically stable when $h<\frac{1}{a}ln\left(\right|b|{\mathrm{e}}^{a\sigma \tau })$.

Proof. 

From scheme (25), we can obtain that

$$y_{k,l}={\left(b{\left(R\left(z\right)\right)}^{\lfloor \sigma m\rfloor }\right)}^k{y}_{0,0}{\left(R\left(z\right)\right)}^l,\forall k\in \mathbb{N},l=0,1,2,\cdots ,m,$$

which implies that the numerical method (25) is asymptotically stable if and only if

$$|b{\left(R\left(z\right)\right)}^{\lfloor \sigma m\rfloor }|<1.$$

(26)

(i) If $a>0$ and s is even,

$$\left|b\right|{\left(R\left(z\right)\right)}^{\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{z\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{ahm\sigma }=\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1,$$

which implies that (26) holds.

(ii) If $a<0$ and r is odd, $h<\frac{1}{a}ln\left(\right|b|{\mathrm{e}}^{a\sigma \tau })$ implies $\left|b\right|{\mathrm{e}}^{a\sigma \tau -ah}<1$. Hence, we can obtain

$$\left|b\right|{\left(R\left(z\right)\right)}^{\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{z\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{ah(m\sigma -1)}=\left|b\right|{\mathrm{e}}^{a\sigma \tau -ah}<1,$$

which implies that (26) holds. □

4.2. Convergence of S2IRKMs

In order to study the convergence of S2IRKM (24), the case where DEDI (1) is defined in the interval $[0,T]$ is considered in this subsection. For convenience, assume that there exists a positive integer N such that $T=N\tau$.

To analyze the local truncation errors of S2IRKM (24) for DEDI (1), consider the following problem:

$$\left\{\begin{array}{cc}{z}_{k,l+1}=z_{k,l}+ah{\displaystyle \sum _{i=1}^v}{b}_i{Z}_{k,l+1}^i,\hfill & l=0,1,\cdots ,m-1,\hfill \\ Z_{k,l+1}^i=z_{k,l}+ah{\displaystyle \sum _{j=1}^v}{a}_{ij}{Z}_{k,l+1}^j,\hfill & i=1,2,\cdots ,v,\hfill \end{array}\right.$$

(27)

where $z_{k,0}=x\left(k{\tau }^{+}\right)$, $z_{k,l}=x\left(t_{k,l}\right)$, $k=0,1,2,·,N$, $l=1,2,·,m-1$.

Theorem 7.

If the corresponding Runge–Kutta method is convergent of order p, then the local truncation errors between (27) and DEDI (1) satisfy that there exists a constant $C_6$ such that for arbitrary $k=0,1,2,·,N$, $l=1,2,·,m-1$,

$$R_{k,l+1}:=|z_{k,l+1}-x\left(t_{k,l+1}\right)|\le C_6{h}^{p+1}.$$

Theorem 8.

If Runge–Kutta methods are convergent of order p, then the impulsive Runge–Kutta methods (24) for (1) are convergent at least of order 1, and in the following sense, there exists a constant $C_{10}$ such that for all $k=0,1,2,\cdots ,N-1$, $l=0,1,2,\cdots ,m$, the global errors satisfy

$$e_{k,l}=|x\left(t_{k,l}\right)-x_{k,l}|\le C_{10}h.$$

Proof. 

From (24) and (27), we have

$$\begin{array}{ccc}\hfill |X_{k,l+1}^i-Z_{k,l+1}^i|& \le & |x_{k,l}-z_{k,l}|+|a|h\left(\sum _{j=1}^v|a_{ij}\left|\right|X_{k,l+1}^i-Z_{k,l}^i|\right)\hfill \\ & \le & |x_{k,l}-z_{k,l}|+|a|h\left(\underset{1\le i\le v}{max}\sum _{j=1}^v\left|a_{ij}\right|\right)\underset{1\le i\le v}{max}\left\{\right|X_{k,l+1}^i-Z_{k,l}^i\left|\right\}\hfill \end{array}$$

which implies

$$\underset{1\le i\le v}{max}\left\{\right|X_{k,l+1}^i-Z_{k,l}^i\left|\right\}\le \Lambda |x_{k,l}-z_{k,l}|$$

where $\Lambda ={\left(1-\left|a\right|h\left({max}_{1\le i\le v}{\sum }_{j=1}^v\left|a_{ij}\right|\right)\right)}^{-1}$. So we can obtain that

$$\begin{array}{ccc}\hfill |x_{k,l+1}-z_{k,l+1}|& \le & |x_{k,l}-z_{k,l}|+|a|h\left(\sum _{i=1}^v\left|b_i\right|\right)\underset{1\le i\le v}{max}\left\{\right|X_{k,l+1}^i-Z_{k,l}^i\left|\right\}\hfill \\ & \le & (1+\beta \Lambda |a\left|h\right)|x_{k,l}-z_{k,l}|,\hfill \end{array}$$

where $\beta =\left({\displaystyle \sum _{i=1}^v}\left|b_i\right|\right)$. By Theorem 7, we have

$$R:=\underset{k=0,1,\cdots ,N-1,l=0,1,\cdots ,m}{max}\left\{R_{k,l}\right\}\le C_6{h}^{p+1}.$$

When $0\le l\le m-1$,

$$\begin{array}{cc}{e}_{k,l+1}\hfill & :=|x\left(t_{k,l+1}\right)-x_{k,l+1}|\hfill \\ & \le |x\left(t_{k,l+1}\right)-z_{k,l+1}|+|z_{k,l+1}-x_{k,l+1}|\hfill \\ \hfill & \le (1+\beta \Lambda |a\left|h\right)|x_{k,l}-z_{k,l}|+R_{k,l+1}\hfill \\ \hfill & \le (1+\beta \Lambda |a\left|h\right)e_{k,l}+R\hfill \\ \hfill & \le {\left(1+\beta \Lambda \left|a\right|h\right)}^{l+1}{e}_{k,0}+\left[{\left(1+\beta \Lambda \left|a\right|{\overline{h}}_1\right)}^{l+1}-1\right]\frac{R}{\beta \Lambda \left|a\right|h}\hfill \\ \hfill & \le {\mathrm{e}}^{(l+1)\beta \Lambda |ah}{e}_{k,0}+\left({\mathrm{e}}^{(l+1)\beta \Lambda \left|a\right|h}-1\right)\frac{R}{\beta \Lambda \left|a\right|h}\hfill \\ \hfill & \le {\mathrm{e}}^{\beta \Lambda |a\tau }{e}_{k,0}+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\tau }-1\right)\frac{R}{\beta \Lambda \left|a\right|h}.\hfill \end{array}$$

(28)

For $\forall k=1,2,\cdots ,N$, from Taylor’s formula, it follows that

$$x\left(r_k\right)-x\left(t_{k,\lfloor \sigma m\rfloor }\right)=x^{\prime }\left(t_{k,\lfloor \sigma m\rfloor }\right)(r_k-t_{k,\lfloor \sigma m\rfloor })+\frac{1}{2!}{x}^{\prime \prime }\left(\xi \right){(r_k-t_{k,\lfloor \sigma m\rfloor })}^2$$

which implies that

$$|x\left(r_k\right)-x\left(t_{k,\lfloor \sigma m\rfloor }\right)|\le C_{7}h,$$

which implies that

$$\begin{array}{ccc}\hfill e_{k+1,0}& =& |x\left(k{\tau }^{+}\right)-x_{k+1,0}|\hfill \\ & =& |bx\left(r_k\right)-b{x}_{k,\lfloor \sigma m\rfloor }|\hfill \\ & \le & \left|b\right|\left(|x\left(r_k\right)-x\left(t_{k,\lfloor \sigma m\rfloor }\right)|+|x\left(t_{k,\lfloor \sigma m\rfloor }\right)-x_{k,\lfloor \sigma m\rfloor }|\right)\hfill \\ & \le & \left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|h\lfloor \sigma m\rfloor }{e}_{k,0}+\left|b\right|\left({\mathrm{e}}^{\beta \Lambda \left|a\right|h\lfloor \sigma m\rfloor }-1\right)\frac{R}{\beta \Lambda \left|a\right|h}+\left|b\right|C_{7}h\hfill \\ & \le & \left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }{e}_{k,0}+\left|b\right|\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R}{\beta \Lambda \left|a\right|h}+\left|b\right|C_{7}h\hfill \\ & \le & \left|b\right|\left(\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\right)\left[\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R}{\beta \Lambda \left|a\right|h}+\left|b\right|C_{7}h\right]\hfill \\ & & +{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}{e}_{0,0}.\hfill \end{array}$$

Because $e_{0,0}=0$, i.e., $x_{0,0}=x\left(0^{+}\right)=x_0$, we have

$$\begin{array}{cc}{e}_{k+1,0}\hfill & \le \left|b\right|\left(\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\right)\left[\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{R}{\beta \Lambda \left|a\right|h}+\left|b\right|C_{7}h\right]\hfill \\ \hfill & \le \left|b\right|\left(\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\right)\left[\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{C_6{h}^p}{\beta \Lambda \left|a\right|}+\left|b\right|C_{7}h\right]\hfill \\ \hfill & \le \left|b\right|\left(\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\right)\left[\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{C_6{T}^{p-1}h}{\beta \Lambda \left|a\right|}+\left|b\right|C_{7}h\right]\hfill \\ \hfill & \le C_{8}h,\hfill \end{array}$$

(29)

where $C_8=\left|b\right|\left(\frac{{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)}^{k+1}-1}{\left(\left|b\right|{\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }\right)-1}\right)\left[\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\sigma \tau }-1\right)\frac{C_6{T}^{p-1}}{\beta \Lambda \left|a\right|}+\left|b\right|C_7\right]$. From (28) and (29), for $k=0,1,\cdots ,N-1$, $l=1,2,\cdots ,m$, we obtain

$$e_{k,l}\le C_{9}h,$$

(30)

where $C_9={\mathrm{e}}^{\beta \Lambda |a\tau }{C}_8+\left({\mathrm{e}}^{\beta \Lambda \left|a\right|\tau }-1\right)\frac{C_6{T}^{p-1}}{\beta \Lambda \left|a\right|}$.

Finally, we know that all the global errors satisfy

$$e_{k,l}\le C_{10}h,\forall k=0,1,2,\cdots ,N-1,\forall l=0,1,2,\cdots ,2m,$$

where $C_{10}=max\{C_8,C_9\}$. □

4.3. Asymptotical Stability of S2I$\theta$Ms

Similarly, the impulsive $\theta$ method for (1) can be constructed as follows:

$$\left\{\begin{array}{cc}{y}_{k,l+1}=y_{k,l}+h(a(1-\theta )y_{k,l}+a\theta y_{k,l+1}),\hfill & l=0,1,\cdots ,m-1,\hfill \\ y_{k+1,0}=b{y}_{k,\lfloor \sigma m\rfloor },\hfill & k\in \mathbb{N},\hfill \\ y_{0,0}=x_0,\hfill \end{array}\right.$$

(31)

Obviously, the impulsive $\theta$ method (31) can be rewritten as

$$\left\{\begin{array}{cc}{y}_{k,l+1}=\left(\frac{1+(1-\theta )ah}{1-\theta ah}\right)y_{k,l},\hfill & l=0,1,\cdots ,m-1,\hfill \\ y_{k+1,0}=b{y}_{k,\lfloor \sigma m\rfloor },\hfill & k\in \mathbb{N},\hfill \\ y_{0,0}=x_0,\hfill \end{array}\right.$$

(32)

Theorem 9. 

Assume the condition (3) holds and $\left|ah\right|<1$:

(i) 

if $a>0$ and $0<\theta <\phi \left(1\right)$, then the impulsive θ method (31) for (1) is asymptotically stable.

(ii) 

if $a<0$ and $0<\theta <\phi \left(0\right)$, then the impulsive θ method (31) for (1) is asymptotically stable when $h<\frac{1}{a}ln\left(\right|b|{\mathrm{e}}^{a\sigma \tau })$.

In the above, $\phi \left(x\right)=\frac{1}{x}-\frac{1}{{\mathrm{e}}^x-1}$, $x\in \mathbb{R}$.

Proof. 

From scheme (32), we can obtain that for $\forall k\in \mathbb{N}$, $0\le l\le m$,

$$y_{k,l}={\left(b{\left(\frac{1+(1-\theta )ah}{1-\theta ah}\right)}^{\lfloor \sigma m\rfloor }\right)}^k{y}_{0,0}{\left(\frac{1+(1-\theta )ah}{1-\theta ah}\right)}^l,$$

Hence, the numerical method (22) is asymptotically stable if and only if

$$\left|b\right|{\left(\frac{1+(1-\theta )a{\overline{h}}_1}{1-\theta a{\overline{h}}_1}\right)}^{\lfloor \sigma m\rfloor }<1.$$

(33)

(i) If $a>0$ and $0<\theta <\phi \left(1\right)$, applying ([21] Lemma 2.3) and the condition (3), we can obtain

$$\left|b\right|{\left(\frac{1+(1-\theta )ah}{1-\theta ah}\right)}^{\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{ah\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{ah\sigma m}=\left|b\right|{\mathrm{e}}^{a\sigma \tau }<1,$$

which implies that (33) holds.

(ii) Similarly, if $a<0$ and $0<\theta <\phi \left(0\right)$, applying ([21] Lemma 2.3) and the condition (3), we can obtain

$$\left|b\right|{\left(\frac{1+(1-\theta )ah}{1-\theta ah}\right)}^{\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{ah\lfloor \sigma m\rfloor }\le \left|b\right|{\mathrm{e}}^{ah(\sigma m-1)}=\left|b\right|{\mathrm{e}}^{a\sigma \tau -ah}<1,$$

which implies that (33) holds. □

5. Numerical Experiments

In this section, two simple numerical examples are given.

Example 3.

Consider the following DEDI:

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=2x\left(t\right),\hfill & t\ge 0,t\ne k,k\in {\mathbb{Z}}^{+},\hfill \\ x\left(k^{+}\right)=\left(\frac{1}{4}\right)x(k-\frac{2}{3}),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0.\hfill \end{array}\right.$$

(34)

Solving (34), we can obtain

$$x\left(t\right)={\left(0.25{\mathrm{e}}^{\frac{2}{3}}\right)}^k{x}_0{\mathrm{e}}^{2(t-k)},\forall t\in (k,k+1].$$

By Theorem 1, the zero solution of (34) is asymptotically stable. By Theorems 2 and 6, both S1IRKM (10) and S2IRKM (24) for (34) are asymptotically stable if the stability function $R_{rs}\left(z\right)=\frac{P_r\left(z\right)}{Q_s\left(z\right)}$ satisfies that s is even. By Theorems 5 and 9, both S1IθM (21) and S2IθM (31) for (34) are asymptotically stable if $0<\theta <\phi \left(1\right)$.

In

Table 1. The errors between the exact solution of DEDI (34) and the numerical solutions obtained from the S1IEM and S2IEM for (34) at $t=6$.

	S1IEM		S2IEM
m	$AE$	$RE$	$AE$	$RE$
100	0.00441823021	0.02184293837	0.01660106964	0.08207271326
200	0.00222814668	0.01101555787	0.01171428183	0.05791331011
400	0.00111888657	0.00553157470	0.00432256363	0.02136997995
800	$0.00056065360$	0.00277177093	0.00300716613	0.01486689045
Ratio	1.98999866598	1.98999866598	1.85487233434	1.85487233434

,

Table 2. The errors between the exact solution of DEDI (34) and the numerical solutions obtained from the S1ICRKM and S2ICRKM for (34) at $t=6$.

	S1ICRKM		S2ICRKM
m	$AE$	$RE$	$AE$	$RE$
100	$8.34915193\times {10}^{-11}$	$4.12767109\times {10}^{-10}$	0.00663129	0.03278391
200	$5.24716381\times {10}^{-12}$	$2.59410376\times {10}^{-11}$	0.00663129	0.03278390
400	$3.28126415\times {10}^{-13}$	$1.62219819\times {10}^{-12}$	0.00167860	0.00829871
800	$2.04836148\times {10}^{-14}$	$1.01267321\times {10}^{-13}$	0.00167860	0.00829871
Ratio	15.97400003	15.97400003	1.98349430	1.98349430

,

Table 3. The errors between the exact solution of DEDI (35) and the numerical solutions obtained from the S1IEM and S2IEM for (35) at $t=10$.

	S1IEM		S2IEM
m	$AE$	$RE$	$AE$	$RE$
100	$2.27936431\times {10}^{-4}$	0.11803342	$1.22261651\times {10}^{-4}$	0.06331134
200	$1.16948119\times {10}^{-4}$	0.06055981	$1.37787810\times {10}^{-4}$	0.07135133
400	$5.92344257\times {10}^{-5}$	0.03067365	$6.32341966\times {10}^{-5}$	0.03274487
800	$2.98092526\times {10}^{-5}$	0.01543627	$2.49977644\times {10}^{-5}$	0.01294471
Ratio	1.97016040	1.97016040	1.86530675	1.86530675

,

Table 4. The errors between the exact solution of DEDI (35) and the numerical solutions obtained from the S1ICRKM and S2ICRKM for (35) at $t=10$.

	S1ICRKM		S2ICRKM
m	$AE$	$RE$	$AE$	$RE$
100	$1.55947583\times {10}^{-11}$	$8.07550914\times {10}^{-9}$	$1.97059664\times {10}^{-4}$	0.10204436
200	$9.68306381\times {10}^{-13}$	$5.01422777\times {10}^{-10}$	$1.38899240\times {10}^{-5}$	0.00719269
400	$6.03074448\times {10}^{-14}$	$3.12292958\times {10}^{-11}$	$1.38899215\times {10}^{-5}$	0.00719269
800	$3.75264056\times {10}^{-15}$	$1.94324801\times {10}^{-12}$	$1.38899213\times {10}^{-5}$	0.00719269
Ratio	16.077341944	16.077341944	5.39574622	5.39574622

and

Table 5. The errors between the exact solution of DEDI (5) and the numerical solutions obtained from the S1ICRKM and S2ICRKM for (5) at $t=10$.

	S1ICRKM		S2ICRKM
m	$AE$	$RE$	$AE$	$RE$
10	$5.96296636\times {10}^{-9}$	$4.56638794\times {10}^{-7}$	$1.90245209\times {10}^{-8}$	$1.45688132\times {10}^{-6}$
20	$3.86900800\times {10}^{-10}$	$2.96285278\times {10}^{-8}$	$1.23953019\times {10}^{-9}$	$9.49221475\times {10}^{-8}$
40	$2.46385915\times {10}^{-11}$	$1.88680198\times {10}^{-9}$	$7.91000980\times {10}^{-11}$	$6.05741693\times {10}^{-9}$
80	$1.55444234\times {10}^{-12}$	$1.19037847\times {10}^{-10}$	$4.99548319\times {10}^{-12}$	$3.82549772\times {10}^{-10}$
Ratio	15.65520355	15.65520355	15.61763152	15.61763152

, AE denotes the absolute errors between the numerical solutions and the exact solutions of DEDIs. Similarly, RE denotes the relative errors between the numerical solutions and the exact solutions of DEDIs.

As can be seen from Table 1, when the step size is halved, both the absolute and relative errors of the scheme 1 impulsive Euler method (S1IEM) and the scheme 2 impulsive Euler method (S2IEM) for DEDI (34) become half of the original ones, which roughly indicates that both the S1IEM and S2IEM for DEDI (34) are convergent of order 1.

As can be seen from Table 2, when the step size is halved, both the absolute and relative errors of the scheme 1 impulsive classical 4-stage 4-order Runge–Kutta method (S1ICRKM) for DEDI (34) become one-sixteenth of the original ones, which roughly indicates that he S1ICRKM for DEDI (34) is convergent of order 4. On the other hand, when the step size is halved, both the absolute and relative errors of the scheme 2 impulsive classical 4-stage 4-order Runge–Kutta method (S2ICRKM) for DEDI (34) become half of the original ones, which roughly indicates that the S2ICRKM for DEDI (34) is convergent of order 1.

Example 4.

Consider the following DEDI:

$$\left\{\begin{array}{cc}{x}^{\prime }\left(t\right)=-2x\left(t\right),\hfill & t\ge 0,t\ne k,k\in {\mathbb{Z}}^{+},\hfill \\ x\left(k^{+}\right)=3x(k-1+\frac{\pi }{4}),\hfill & k\in {\mathbb{Z}}^{+},\hfill \\ x(0+0)=x_0.\hfill \end{array}\right.$$

(35)

Solving (34), we can obtain

$$x\left(t\right)={\left(3{\mathrm{e}}^{-\frac{\pi }{2}}\right)}^k{x}_0{\mathrm{e}}^{-2(t-k)},\forall t\in (k,k+1].$$

By Theorem 1, the zero solution of (35) is asymptotically stable. By Theorems 2 and 6, both S1IRKM (10) and S2IRKM (24) for (35) are asymptotically stable if the stability function $R_{rs}\left(z\right)$ satisfies that r is odd. By Theorems 5 and 9, both S1IθM (21) and S2IθM (31) for (35) are asymptotically stable if $0<\theta <\phi \left(0\right)=0.5$.

As can be seen from Table 3, when the step size is halved, both the absolute and relative errors of the S1IEM and S2IEM for DEDI (35) become half of the original ones, which roughly indicates that both the S1IEM and S2IEM for DEDI (35) are convergent of order 1.

As can be seen from Table 4, when the step size is halved, both the absolute and relative errors of the S1ICRKM for DEDI (35) become one-sixteenth of the original ones, which roughly indicates that the S1ICRKM for DEDI (35) is convergent of order 4. On the other hand, when the step size is halved, both the absolute and the relative errors of the S2ICRKM for DEDI (34) become one-fifth of the original ones, which roughly indicates that the S2ICRKM for DEDI (35) is convergent but not up to order 4.

As can be seen from Table 5, which is different from Table 4 and Table 4, the S2ICRKM for DEDI (5) is convergent of order 4 when the step sizes satisfy $h=\frac{\tau }{m}$ and $m\sigma =\lfloor m\sigma \rfloor$, i.e., $r_k=t_{k,\lfloor m\sigma \rfloor }$, $k=0,1,2,\cdots$.

6. Conclusions and Future Works

In this paper, two different schemes of impulsive Runge–Kutta methods are constructed for DEDI (1) based on different ways to approximate the states $x\left(r_k\right)$, where $k\in {\mathbb{Z}}^{+}$ is required for the delayed impulses. When constructing S1IRKMs, the approximations of $x\left(r_k\right)$ are the numerical solutions obtained from Runge–Kutta methods at moments $r_k$, $k\in {\mathbb{Z}}^{+}$. The S1IRKMs have better convergence and are convergent of order p if the corresponding Runge–Kutta method is p order. On the other hand, when constructing S2IRKMs, the approximations of $x\left(r_k\right)$ are the numerical solutions obtained from Runge–Kutta methods at moments at $t_{k,\lfloor \sigma m\rfloor }$, where $t_{k,\lfloor \sigma m\rfloor }\le r_k$, $|r_k-t_{k,\lfloor \sigma m\rfloor }|\le h$, $h=\frac{\tau }{m}$, $m,k\in {\mathbb{Z}}^{+}$. The S2IRKMs in the general case are only convergent of order 1, but they are more concise and may suit for more complex differential equations with delayed impulses. Therefore, it is very necessary to consider S2I$\theta$M. Moreover, the asymptotical stability of the exact solution and the numerical solutions of DEDI (1) was studied.

Here, we only studied the asymptotical stability of the exact solution of linear DEDI (1); the asymptotical stability of the exact solution of nonlinear DEDIs still needs further study. Moreover, applying S2I$\theta$Ms to solve nonlinear DEDIs, time-delay differential equations with delayed impulses, and stochastic differential equations with delayed impulses will be future work. Applying impulsive continuous Runge–Kutta methods to solve these equations will also be future work.