Dissipativity of extended Pouzet–Runge–Kutta methods for neutral delay integro-differential equations

References

• K. Burrage and J.C. Butcher, Nonlinear stability of a general class of differential equation methods, BIT 20 (1980), pp. 185–203. doi: 10.1007/BF01933191
   Google Scholar
• S.Q. Gan, Dissipativity of linear θ-methods for integro-differential equations, Int. J. Comput. Math. Appl. 52 (2006), pp. 449–458. doi: 10.1016/j.camwa.2006.02.010
   Web of Science ®Google Scholar
• S.Q. Gan, Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations, J. Comput. Appl. Math. 206 (2007), pp. 898–907. doi: 10.1016/j.cam.2006.08.030
   Web of Science ®Google Scholar
• S.Q. Gan, Dissipativity of θ-methods for nonlinear delay differential equations of neutral type, Appl. Numer. Math. 59 (2009), pp. 1354–1365. doi: 10.1016/j.apnum.2008.08.003
   Web of Science ®Google Scholar
• A.T. Hill, Dissipativity of Runge–Kutta methods in Hilbert spaces, BIT 37a (1997), pp. 37–42. doi: 10.1007/BF02510171
   Google Scholar
• A.T. Hill, Global dissipativity for A-Stable methods, SIAM J. Numer. Anal. 34 (1997), pp. 119–142. doi: 10.1137/S0036142994270971
   Web of Science ®Google Scholar
• C.M. Huang, Dissipativity of Runge–Kutta methods for dynamical systems with delays, IMA J. Numer. Anal. 20 (2000), pp. 153–166. doi: 10.1093/imanum/20.1.153
   Web of Science ®Google Scholar
• C.M. Huang, Dissipativity of one-leg methods for dynamical systems with delays, Appl. Numer. Math. 35 (2000), pp. 11–22. doi: 10.1016/S0168-9274(99)00048-3
   Web of Science ®Google Scholar
• C.M. Huang, Dissipativity of multistep Runge–Kutta methods for dynamical systems with delays, Math. Comput. Modelling 40 (2004), pp. 1285–1296. doi: 10.1016/j.mcm.2005.01.019
   Web of Science ®Google Scholar
• C.M. Huang and G.N. Chen, Dissipativity of linear θ-method for dynamical systems with delays, Math. Numer. Sin. 22 (2000), pp. 501–506 (in Chinese).
   Google Scholar
• A.R. Humphries and A.M. Stuart, Runge–Kutta methods for dissipativity and gradient dynamical systems, SIAM J. Numer. Anal. 31 (1994), pp. 1452–1485. doi: 10.1137/0731075
   Web of Science ®Google Scholar
• A.R. Humphries and A.M. Stuart, Model problems in numerical stability theory for initial value problems, SIAM Rev. 36 (1994), pp. 226–257. doi: 10.1137/1036054
   Web of Science ®Google Scholar
• X.Y. Liu and L.P. Wen, Dissipativity of one-leg methods for neutral delay integro-differential equations, J. Comp. Appl. Math. 225 (2010), pp. 165–173. doi: 10.1016/j.cam.2010.05.030
   Web of Science ®Google Scholar
• R. Qi, C.J. Zhang, and Y.J. Zhang, Dissipativity of multistep Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations, Acta Math. Appl. Sin. 28(2) (2012), pp. 225–236. doi: 10.1007/s10255-012-0142-x
   Web of Science ®Google Scholar
• H.J. Tian, Numerical and analytic dissipativity of the θ-methods for delay differential equations with a bounded lag, Int. J. Bifur. Chaos 14 (2004), pp. 1839–1845. doi: 10.1142/S0218127404010096
   Web of Science ®Google Scholar
• H.J. Tian, L.Q. Fan, and J.X. Xiang, Numerical dissipativity of multistep methods for delay differential equations, Appl. Math. Comput. 188 (2007), pp. 934–941. doi: 10.1016/j.amc.2006.10.048
   Web of Science ®Google Scholar
• H.J. Tian, N. Guo, and A. Shen, Dissipativity of delay functional differential equations with bounded lag, J. Math. Anal. Appl. 355 (2009), pp. 778–782. doi: 10.1016/j.jmaa.2009.02.024
   Web of Science ®Google Scholar
• L.P. Wen, W.S. Wang, and Y.X. Yu, Dissipativity of θ-methods for a class of nonlinear neutral delay differential equations, J. Math. Anal. Appl. 347 (2008), pp. 780–786. doi: 10.1016/j.jmaa.2008.05.007
   Web of Science ®Google Scholar
• L.P. Wen, W.S. Wang, and Y.X. Yu, Dissipativity of Runge–Kutta methods for neutral delay-integro-differential equations, Appl. Math. Comput. 215 (2009), pp. 583–590. doi: 10.1016/j.amc.2009.05.039
   Web of Science ®Google Scholar
• L.P. Wen, W.S. Wang, and Y.X. Yu, Dissipativity and asymptotic stability of nonlinear neutral delay-integro-differential equations, Nonlinear Anal. 72 (2010), pp. 1746–1754. doi: 10.1016/j.na.2009.09.016
   Web of Science ®Google Scholar
• A.G. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces, Math. Numer. Sin. 22 (2000), pp. 429–436 (in Chinese).
   Google Scholar
• C.J. Zhang and S. Vandewalle, Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations, IMA J. Numer. Anal. 24 (2004), pp. 193–214. doi: 10.1093/imanum/24.2.193
   Web of Science ®Google Scholar
• C.J. Zhang, T.T. Qin, and J. Jin, The extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations, Computing 90 (2010), pp. 57–71. doi: 10.1007/s00607-010-0103-2
   Web of Science ®Google Scholar