Dissipativity Of Numerical Methods For Neutral Delay Differential Equations With Piecewise Constant Delay

Neutral delay differential equations with piecewise constant delay have been widelyapplied to science and engineering, such as biology, economics, electrostatics, hydrody-namic, electromagnetic field theory, chemistry and control theory and so on. It is importantfor studying the theory and numerical methods for neutral delay differential equations withpiecewise constant delay. Dissipativity is an important property of dynamical systems. Whatis called Dissipativity of a dynamical systems means that the system possess a bounded ab-sorbing set that all trajectories enter in a finite time and thereafter remain inside. Such asthe Lorenz equations and two dimensional Navier-Stokes equation and many systems are alldissipative. When used the numerical methods to solve these systems, we hope numericalmethods that retain the dissipativity of the system.In2008, Wansheng Wang and Shoufu Li studied the dissiptivity of the neutral delaydifferential equations with piecewise constant delay itself and Runge-Kutta methods to solvethese systems (Dissipativity of Runge-Kutta methods for nonlinear neutral delay differentialequations with piecewise constant delay, Appl. Math. Lett.21(2008)983-991).In this paper, at first, we studies dissipativity of a class of linear multistep methodsfor solving the problems mentioned above, and the sufficient conditions are given whichguarantees the methods inherit the dissipativity of system itself.Secondly, we further extend the problem class up to multiple delay problems, andstudy dissipativity of this kind of multiple delay problems system itself, and two classesof sufficient conditions are given in order to keep the problems is dissipative.Finally, the dissipativity of Runge-Kutta method for solving the multiple delay problemis studied and two classes of sufficient conditions are given in order to guarantee the methodsto be dissipative.Some numerical experiments are present, the results further verify the theoretical re-sults.