Numerical Stability Of Delay-Differential-Algebraic Systems And Block Methods

Delay differential algebraic equations (DDAEs), which have both delay and algebraic constraints, arise in a wide variety of scientific and engineering applications, including circuit analysis, computer-aided design and real-time simulation of mechanical (multibody) systems, chemical process simulation and optimal control. However, because of the complexity of the DDAEs systems, it becomes quite difficult to obtain the analytic solutions. Therefore, it is necessary to investigate the numerical methods for DDAEs.The numerical solution of differential algebraic equations (DAEs) has been the subject of intense research activity in the past few years. A great deal of progress has been made in the analysis of numerical methods, the development of efficient mathematical software implementing the numerical methods. During the same period of time, much work has also been done in the field of numerical solution of delay differential equations (DDEs). The stability and convergence of numerical methods for DDEs has been very intensively studied. For the moment, there are only few researches on the numerical solution of DDAEs. This paper applies block methods and multistep Runge-Kutta methods to systems of linear delay differential algebraic equations. For A-stable block methods and multistep Runge-Kutta methods with some conditions, the numerical solution is asymptotically stable. This paper investigates the block implicit one-step methods for index 1 and 2 Hessenberg DDAEs. The error estimation of the algorithms is also given.For delay differential equations, especially for state-dependent delay differential equations, jump discontinuities can occur in various derivatives of the solution. For these discontinuities, the implementation of the algorithm is discussed. The software implementing block methods and multistep Runge-Kutta methods are designed, and have been as parts of software packs for delay systems. A number of numerical examples are also presented for some problems. Numerical experiments show that these numerical algorithms are efficient.