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\(H_{\alpha}\)-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
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References:

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