## $$H_{\alpha}$$-stability of modified Runge-Kutta methods for nonlinear neutral pantograph equations.(English)Zbl 1125.65076

The authors investigate the $$H_\alpha$$-stability of algebraically stable Runge-Kutta methods with a variable stepsize for the following system
$\begin{cases} y'(t)=f(t,y(t))+g(t, y(qt))+c(t)y'(qt),& t>0,\\ y(0)=y_0,\end{cases} \tag{1}$
where $$f,g:[0,\infty)\times C^d\to C^d$$ are continuous and $$c(t)$$ is a continuous $$d\times d$$-matrix function in $$[0,\infty)$$. The first section is an introduction concerning the nonlinear neutral pantograph equation:
$\begin{cases} y'(t)=F(t,(t),y(qt),y'(qt)),& t>0,\\ y(0)=y_0,\end{cases}\tag{2}$
where $$0<q<1$$, $$y_0\in C^d$$, $$F:\mathbb R^+\times C^d \times C^d\times C^d\to C^d$$ continuous, and the stability properties of the appropriate numerical methods used for the numerical resolution of this kind of equations. The second section focuses on the modified Runge-Kutta method $$(A,b, c)$$ with the form:
\begin{aligned} y_{n+1} & =y_n+h_n \sum^s_{i=1}b_if(t^i_n,y_i^{n+1}), \\ y_i^{n+1} & =y_n+\overline {h_n}\sum^s_{j=1}a_{ij}f(t^j_n,y_j^{n+1}),\quad i=1,2, \dots,s\end{aligned}
where $$y_0\in C^d$$, $$f:[0,\infty)\times C^d\to C^d$$ is a continuous function, $$\Delta=\{0=t_0<t_1<\cdots<t_n=T>0\}$$ represents a mesh, $$h_{n+1}=t_{n+1}-t_n$$ the stepsize, $$t^i_n=t_n+ c_ih_n$$ and $$\overline{h_n}=(1+ \alpha_n(h_n))h_n$$ with $$\alpha_n (\eta)$$ such that
$\begin{cases} \alpha_n(\eta)=O (\eta^p) & \text{ as }\eta \to 0,\\ \alpha_n(\eta)>0, & \text{ for all }\eta,\end{cases}$
Asymptotical stability conditions of the analytic solutions of (1) are presented. The third section is devoted to the stability analysis of the modified Runge-Kutta method, giving the conditions for the $$H_\alpha$$-stability. It points out the $$H_\alpha$$-stability for the Radau IA, Radau IIA and Lobato IIIC methods, the odd-stage Gauss-Legendre methods and the one-leg $$\theta$$ methods with $$\tfrac 12\leq\theta\leq 1$$. In the last section one gives two numerical experiments are presented: the linear neutral pantograph equation and the nonlinear neutral pantograph equation.

### MSC:

 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K40 Neutral functional-differential equations
Full Text:

### References:

  Bellen, A.; Guglielmi, N.; Torelli, L., Asymptotic stability properties of θ-methods for the pantograph equation, Appl. numer. math., 24, 279-293, (1997) · Zbl 0878.65064  Dekker, K.; Verwer, J.G., Stability of runge – kutta methods for stiff nonlinear differential equations, (1984), North-Holland Amsterdam · Zbl 0571.65057  Derfel, G.A., Kato problem for functional equations and difference Schrödinger operators, J. operator theory, 46, 319-321, (1990) · Zbl 0728.34067  Fox, L.; Mayers, D.F.; Ockendon, J.R.; Tayler, A.B., On a functional differential equation, J. inst. math. appl., 8, 271-307, (1971) · Zbl 0251.34045  Guglielmi, N.; Zennaro, M., Stability of one-leg θ-methods for the variable coefficient pantograph equation on the quasi-geometric mesh, IMA J. numer. anal., 23, 421-438, (2003) · Zbl 1055.65094  Huang, Cheng Ming; Fu, Hong Yuan; Li, Shou Fu; Chen, Guang Nan, Stability analysis of runge – kutta methods for non-linear delay differential equations, Bit, 39, 270-280, (1999)  Huang, Cheng Ming; Vandewalle, Stefan, Discretized stability and error growth of the nonautonomous pantograph equation, SIAM J. numer. anal., 42, 2020-2024, (2005) · Zbl 1080.65068  Iserles, A.; Terjekj, J., Stability and asymptotic stability of functional – differential equations, J. London math. soc. (2), 51, 559-572, (1995) · Zbl 0832.34080  Koto, T., Stability of runge – kutta methods for the generalized pantograph equation, Numer. math., 84, 870-884, (1999) · Zbl 0890.65088  Liang, J.; Liu, M.Z., Numerical stability of θ-methods for pantograph delay differential equations, J. numer. methods comput. appl., 12, 271-278, (1996), (in Chinese)  Liu, M.Z.; Yang, Z.W.; Xu, Y., The stability of modified runge – kutta methods for pantograph equation, Math. comp., 75, 1201-1215, (2006) · Zbl 1094.65075  Liu, Y., Stability analysis of θ-methods for delay differential equations with infinite lag, J. comput. appl. math., 71, 177-190, (1996) · Zbl 0853.65076  Liu, Y., Numerical investigation of the pantograph equation, Appl. numer. math., 24, 309-317, (1997) · Zbl 0878.65065  Qiu, Lin; Yang, Biao; Kuang, Jiao Xun, The NGP-stability of runge – kutta methods for systems of neutral delay differential equations, Numer. math., 81, 451-459, (1999) · Zbl 0918.65061  Xu, Y.; Liu, M.Z., $$\mathcal{H}$$-stability of runge – kutta methods with general variable stepsize for pantograph equation, Appl. math. comput., 148, 881-892, (2004) · Zbl 1038.65070  Zhang, C.J.; Sun, G., The discrete dynamics of nonlinear infinite-delay-differential equations, Appl. math. lett., 15, 521-526, (2002) · Zbl 1001.65091  Zhang, C.J.; Sun, G., Nonlinear stability of runge – kutta methods applied to infinite-delay-differential equations, Math. comput. modelling, 39, 495-503, (2004) · Zbl 1068.65106  Zhang, C.J.; Zhou, S.Z., Nonlinear stability and D-convergence of runge – kutta methods for delay differential equations, J. comput. appl. math., 85, 225-237, (1997) · Zbl 0904.65082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.