## On the stability of continuous quadrature rules for differential equations with several constant delays.(English)Zbl 0771.65053

This paper is concerned with the stability of (non-confluent) Runge-Kutta methods for solving the pure delay differential equation $y'(t)=f(t,y(t- \tau),y(t-2\tau),\dots,y(t-R\tau)),\;t>0,$ where $$\tau>0$$ is a constant delay and $$y(t)=\varphi(t)$$ on $$[-R\tau,0]$$. The method is described by a Butcher array and an appropriate natural continuous extension; it is shown that this gives rise to the same continuous quadrature rule regardless of the choice of the matrix $$A$$. The stability analysis is based on the test equation $y'(t)=-\sum^ R_{r=1}b_ r(t)y(t-r\tau), \tag{1}$ where the $$b_ r$$ are nonnegative (smooth) functions. If $$\beta(t):=\sum^ R_{r=1}b_ r(t)$$ satisfies $\int^{x+R\tau}_ x\beta(s)ds\leq 1\quad\text{for all } x\geq 0$ and $$\beta(t)>0$$ for every $$t\geq R\tau$$ then $$| y(t)|\leq 2\max\{|\varphi(x)|:- R\tau\leq x\leq 0\}$$, $$t\geq 0$$, holds for every solution $$y(t)$$ of (1). Thus, if the given method, when applied to (1), generates a numerical solution $$\{y_ n\}$$ (on a uniform, constrained mesh with $$h=\tau/N$$ $$(N\in\mathbb{N}))$$ which satisfies the analogous inequality $$| y_ n|\leq 2\max\{|\varphi(t)|:-R\tau\leq t\leq 0\}$$ whenever $$\beta(t)\leq 1/(R\tau)$$ $$(t\geq 0)$$, then it is said to be $$QN_ 0$$- stable. The authors characterize $$QN_ 0$$-stable continuous quadrature rules and present several concrete examples of such methods.

### MSC:

 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 34K05 General theory of functional-differential equations
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