Regularity properties of Runge-Kutta methods for delay differential equations
The authors extend well-known results on Runge-Kutta methods for ordinary differential equations to delay-differential equations. Consider the initial value problem with constant delay \(\tau>0\) \[ y'(t)= g(y(t),y(t-\tau)) \quad\text{for }t\geq t_0,\qquad y(t)= \varphi(t)\quad\text{for }t\in[t_0-\tau,t_0] \tag{1} \] with continuous functions \(g:\mathbb{R}^q\times \mathbb{R}^q\to\mathbb{R}^q\) and a given initial function \(\varphi\in C([t_0-\tau,t_0],\mathbb{R}^q)\). To solve (1) consider Runge-Kutta methods of the form \[ \begin{aligned} Y_i^{n+1}&= y_h(t_n)+h \sum_{j=1}^s a_{ij} g(Y_j^{n+1}, y_h(t_{n-m}+c_jh)),\\ y_h(t_n+\theta h)&= y_h(t_n)+h \sum_{j=1}^s b_j(\theta)\cdot g(Y_j^{n+1}, y_h(t_{n-m}+ c_jh)), \end{aligned}\tag{2} \] \(i=1,2,\dots, s\), where \(s\) is the number of stages, \(n\in\mathbb{N}_0\), \(\theta\in[0,1]\), \(h\) is a constant stepsize such that \(mh=\tau\) for some integer \(m>0\), \(y_h\) is the numerical approximation to the solution \(y\) of (1) at the gridpoints and \(b_j(\theta)\) are polynomials. Now regularity means that if the numerical approximations \(y_h(t_n)\) at the gridpoints \(t_n\) converge to \(\overline{y}\) as \(n\to\infty\), then \(\overline{y}\in \{y\in\mathbb{R}^q\), \(g(y,y)=0\}\), i.e. Runge-Kutta methods (2) cannot have spurious asymptotics. Strong regularity means that not only \(y_h(t_n)\), but also the stage values \(Y_i^{n+1}\) obtained by interpolation, tend to the equilibrium point of (1) as \(n\to\infty\). The authors derive conditions which characterize regular and strongly regular Runge-Kutta methods for systems (1). These conditions are then used to derive examples of strongly regular methods with \(s=p=2\) and \(s=p=3\).
scientific article; zbMATH DE number 1036658