Runge–Kutta Methods for Systems of Differential Equation with Piecewise Continuous Arguments: Convergence and Stability. (19th May 2018)

• Record Type: Journal Article
• Authors: Wang, Qi; Wang, Xiaoming
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 7(2018)
• Page Start: 784
• Page End: 799
• Publication Date: 2018-05-19
• Subjects: Convergence -- piecewise continuous arguments -- Runge–Kutta methods -- stability; 65L07 -- 65L20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2017.1421554
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: This work deals with the convergence and stability of Runge–Kutta methods for systems of differential equation with piecewise continuous arguments x ′ ( t ) = Px ( t )+ Qx ([ t +1∕2]) under two cases for coefficient matrix. First, when P and Q are complex matrices, the sufficient condition under which the analytic solution is asymptotically stable is given. It is proven that the Runge–Kutta methods are convergent with order p . Moreover, the sufficient condition under which the analytical stability region is contained in the numerical stability region is obtained. Second, when P and Q are commutable Hermitian matrices, using the theory of characteristic, the necessary and sufficient conditions under which the analytic solution and the numerical solution are asymptotically stable are presented, respectively. Furthermore, whether the Runge–Kutta methods preserve the stability of analytic solution are investigated by the theory of Padé approximation and order star. To demonstrate the theoretical results, some numerical experiments are adopted.