Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations. (English) Zbl 1018.65101

Zhang, Chengjian

J. Comput. Math. 20, No. 6, 583-590 (2002).

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.

Reviewer: Hermann Brunner (St.John’s)

MSC:

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

Keywords:

nonlinear stability; Runge-Kutta methods; systems of nonlinear neutral delay differential equations; asymptotic stability

Citations:

Zbl 0887.65093