Stability analysis of linear multistep and Runge-Kutta methods for neutral multidelay-differential-algebraic systems

Abstract This paper is concerned with the stability analysis of linear multistep and Runge–Kutta methods for neutral multidelay-differential-algebraic systems. We focus on the system with no restrictive assumption on the structure of the coefficient matrices. It is shown that under some conditions, A -stable linear multistep methods and A -stable Runge–Kutta methods can preserve the delay-independent stability of the underlying linear systems. Numerical experiments further confirm the theoretical results.

[1]  Chengjian Zhang,et al.  The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations , 1998 .

[2]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[3]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[4]  Hao Chen,et al.  Asymptotic stability of block boundary value methods for delay differential-algebraic equations , 2010, Math. Comput. Simul..

[5]  Chengjian Zhang,et al.  Stability analysis of LMMs for systems of neutral multidelay-differential equations , 1999 .

[6]  Stephen L. Campbell,et al.  Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions , 2009, Appl. Math. Comput..

[7]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[8]  Robert Schaback,et al.  Solvability of partial differential equations by meshless kernel methods , 2008, Adv. Comput. Math..

[9]  Linda R. Petzold,et al.  Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type , 1998 .

[10]  Uri M. Ascher,et al.  The numerical solution of delay-differential-algebraic equations of retarded and neutral type , 1995 .

[11]  Uri M. Ascher,et al.  Stability of Computational Methods for Constrained Dynamics Systems , 1991, SIAM J. Sci. Comput..

[12]  Stefan Vandewalle,et al.  Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays , 2009 .

[13]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[14]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[15]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[16]  Linda R. Petzold,et al.  Asymptotic stability of linear delay differential-algebraic equations and numerical methods , 1997 .

[17]  Chengjian Zhang,et al.  Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations , 2008, Adv. Comput. Math..

[18]  M. Z. Liu,et al.  Stability of the Rosenbrock methods for the neutral delay differential-algebraic equations , 2005, Appl. Math. Comput..