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Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations. (English) Zbl 1018.65101
T. Koto [Japan J. Ind. Appl. Math. 14, No. 1, 111-123 (1997; Zbl 0887.65093)] adapted natural Runge-Kutta methods to systems of nonlinear neutral delay differential equations and analyzed their asymptotic stability in $$\mathbb{R}^d$$, using a discrete Lyapunov functional. In the present paper an alternative approach extends this analysis to systems of the form ${d\over dt} [y(t)- Ny(t- \tau)]= f(t, y(t), y(t-\tau)),\quad t\geq 0\;(\tau> 0),\tag{1}$ where $$N\in\mathbb{C}^{d\times d}$$ is a constant (complex) matrix with $$\|N\|< 1$$. It is shown that a natural Runge-Kutta method for (1) based on a $$(k,l)$$-algebraically stable Runge-Kutta method for ordinary differential equations inherits, under certain conditions, the asymptotic stability properties of the original method.

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