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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.

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)