Wang, Wansheng; Li, Dongfang Stability analysis of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations. (English) Zbl 1265.65276 Numer. Math., Theory Methods Appl. 4, No. 4, 537-561 (2011). Summary: This paper is concerned with the numerical stability of implicit Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay. Using a Halanay inequality generalized by E. Liz and S. Trofimchuk [J. Math. Anal. Appl. 248, No. 2, 625–644 (2000; Zbl 0965.34063)], we give two sufficient conditions for the stability of the true solution to this class of equations. Runge-Kutta methods with compound quadrature rule are considered. Nonlinear stability conditions for the proposed methods are derived. As an illustration of the application of these investigations, the asymptotic stability of the presented methods for Volterra delay-integro-differential equations is proved under some weaker conditions than those in literatures. An extension of the stability results to such equations with weakly singular kernel is also discussed. Cited in 4 Documents MSC: 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations 45G05 Singular nonlinear integral equations Keywords:Runge-Kutta methods; stability; nonlinear neutral Volterra delay-integro-differential equations; weakly singular kernel Citations:Zbl 0965.34063 Software:FODE PDFBibTeX XMLCite \textit{W. Wang} and \textit{D. Li}, Numer. Math., Theory Methods Appl. 4, No. 4, 537--561 (2011; Zbl 1265.65276) Full Text: DOI