Stability Analysis of Additive Runge-Kutta Methods for Delay-Integro-Differential Equations

. This paper is concerned with stability analysis of additive Runge-Kutta methods for delay-integro-differential equations. We show that if the additive Runge-Kutta methods are algebraically stable, the perturbations of the numerical solutions are controlled by the initial perturbations from the system and the methods.


Introduction
Spatial discretization of many nonlinear parabolic problems usually gives a class of ordinary differential equations, which have the stiff part and the nonstiff part; see, e.g., [1][2][3][4][5].In such cases, the most widely used time-discretizations are the special organized numerical methods, such as the implicitexplicit numerical methods [6,7], the additive Runge-Kutta methods [8][9][10][11][12], and the linearized methods [13,14].When applying the split numerical methods to numerically solve the equations, it is important to investigate the stability of the numerical methods.
The investigation on stability analysis of different numerical methods for problem (1) has fascinated generations of researchers.For example, Torelli [15] considered stability of Euler methods for the nonautonomous nonlinear delay differential equations.Hout [16] studied the stability of Runge-Kutta methods for systems of delay differential equations.Baker and Ford [17] discussed stability of continuous Runge-Kutta methods for integrodifferential systems with unbounded delays.Zhang and Vandewalle [18] discussed the stability of the general linear methods for integrodifferential equations with memory.Li and Zhang obtained the stability and convergence of the discontinuous Galerkin methods for nonlinear delay differential equations [19,20].More references for this topic can be found in [21][22][23][24][25][26][27][28][29][30].However, few works have been found on the stability of splitting methods for the proposed methods.
In the present work, we present the additive Runge-Kutta methods with some appropriate quadrature rules 2 International Journal of Differential Equations to numerically solve the nonlinear delay-integrodifferential equations (1).It is shown that if the additive Runge-Kutta methods are algebraically stable, the obtained numerical solutions are globally and asymptotically stable under the given assumptions, respectively.The rest of the paper is organized as follows.In Section 2, we present the numerical methods for problems (1).In Section 3, we consider stability analysis of the numerical schemes.Finally, we present some extensions in Section 4.

The Numerical Methods
In this section, we present the additive Runge-Kutta methods with the appropriate quadrature rules to numerically solve problem (1).
The coefficients of the additive Runge-Kutta methods can be organized in Buther tableau as follows (cf.[31]):

Theorem 2. Assume an additive Runge-Kutta method is algebraically stable and
where   and   are numerical approximations to problems (1) and ( 5), respectively.
Remark 4. In [35], Yuan et al. also discussed nonlinear stability of additive Runge-Kutta methods for multidelayintegro-differential equations.However, the main results are different.The main reason is that the results in [35] imply that the perturbations of the numerical solutions tend to infinity when the time increase, while the stability results in present paper indicate that the perturbations of the numerical solutions are independent of the time.Besides, the asymptotical stability of the methods is also discussed in the present paper.

Conclusion
The additive Runge-Kutta methods with some appropriate quadrature rules are applied to solve the delay-integrodifferential equations.It is shown that if the additive Runge-Kutta methods are algebraically stable, the obtained numerical solutions can be globally and asymptotically stable, respectively.In the future works, we will apply the methods to solve more real-world problems.