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

Torelli, L.; Vermiglio, R.

IMA J. Numer. Anal. 13, No. 2, 291-302 (1993).

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.

Reviewer: H.Brunner (St.John’s)

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

Keywords:

multiple delays; stability; Runge-Kutta methods; delay differential equation; Butcher array; continuous quadrature rule; test equation