Stability regions of one step mixed collocation methods for y″=f(x,y)

We analyze the linear stability properties of mixed collocation Runge-Kutta-Nystrom (RKN) methods for the second order Ordinary Differential Equations y'' = f(x, y). In particular we consider the influence of the collocation nodes on the stability region, analyzing in detail the case of frequency and step length dependent RKN methods obtained through mixed collocation which are based on Gauss, Radau and Lobatto nodes.

[1]  Ben P. Sommeijer,et al.  Diagonally implicit Runge-Kutta-Nystrm methods for oscillatory problems , 1989 .

[2]  Beatrice Paternoster,et al.  Stability analysis of frequency and step length dependent Runge-Kutta-Nyström methods , 2006, Future Gener. Comput. Syst..

[3]  Beatrice Paternoster,et al.  A conditionally P-stable fourth-order exponential-fitting method for y '' = f ( f, y ) , 1999 .

[4]  Beatrice Paternoster,et al.  Analysis of Stability of Rational Approximations through Computer Algebra , 1999, CASC.

[5]  Linda R. Petzold,et al.  Numerical solution of highly oscillatory ordinary differential equations , 1997, Acta Numerica.

[6]  Beatrice Paternoster,et al.  Computation of the Interval of Stability of Runge-Kutta-Nyström Methods , 1998, J. Symb. Comput..

[7]  Beatrice Paternoster,et al.  Runge-Kutta(-Nystro¨m) methods for ODEs with periodic solutions based on trigonometric polynomials , 1998 .

[8]  Nguyen Huu Cong,et al.  Stability of collocation-based Runge-Kutta-Nyström methods , 1991 .

[9]  L. Kramarz.  Stability of collocation methods for the numerical solution ofy″=f (x,y) , 1980 .

[10]  Liviu Gr. Ixaru,et al.  P-stability and exponential-fitting methods for y″″ = f(x, y) , 1996 .

[11]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[12]  John P. Coleman,et al.  Mixed collocation methods for y ′′ =f x,y , 2000 .

[13]  Beatrice Paternoster,et al.  A note on the capacitance matrix algorithm, substructuring, and mixed or Neumann boundary conditions , 1987 .

[14]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[15]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[16]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .