Stabilities of (A,B,C) and NPDIRK methods for systems of neutral delay-differential equations with multiple delays. (English) Zbl 1043.65091
The authors consider the following neutral delay-differential equations with multiple delays (NMDDE)
\[
y'(t)=Ly(t)+\sum^m_{j=1}[M_j y(t-\tau_j)+N_j y'(t-\tau_j)], \quad t\geq 0, \tag{\(*\)}
\]
where \(\tau>0, L, M_j\) and \(N_j\) are constant complex-value \(d\times d\) matrices. A sufficient condition for the asymptotic stability of NMDDE system (\(*\)) is given. The stability of Butcher’s (A,B,C)-method for systems of NMDDE are studied. In addition, they present a parallel diagonally-implicit iteration Runge-Kutta (PDIRK) method (NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit Runge-Kutta methods. The stability of a special class of NPDIRK methods is investigated by analyzing their stability behaviours of the solutions of (\(*\)). Numerical experiments have not be performed.
Reviewer: Guido Vanden Berghe (Gent)
MSC:
65L20 | Stability and convergence of numerical methods for ordinary differential equations |
34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |
65Y05 | Parallel numerical computation |
65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |