STABILITIES OF （A,B,C）AND NPDIRK METHODS FOR SYSTEMS OF NEUTRAL DELAY—DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

Stabilities of (A,B,C) and NPDIRK Methods for Systems of Neutral Delay-Differential Equations with Multiple Delays

Consider the following neutral delay-differential equations with multiple delays (NMDDE)$$y'(t)=Ly(t)+sum_{j=1}^{m}[M_jy(t-tau_j)+N_jy'(t-tau_j)], tgeq 0, (0.1)$$ where $tau>0$, $L, M_j$ and $N_j$ are constant complex- value $d×d$ matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE is studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK) methods (NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit RK methos. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).