P-stable Mono-Implicit Runge-Kutta-Nyström Modifications of the Numerov Method

We present families of fourth-order mono-implicit Runge-Kutta-Nystrom methods. Each member of these families can be considered as a modification of the Numerov method. Some parameters of these new methods are used to optimize the linear stability properties, i.e. to obtain P-stable methods with a minimal phase-lag. Also we show that in some cases there exist P-stable methods with stage-order 3. Since the methods considered are mono-implicit, the computational work needed in each time-step to solve the implicit equations is reduced seriously.

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

[2]  T. E. Simos,et al.  Numerical integration of the one-dimensional Schro¨dinger equations , 1990 .

[3]  Jeff Cash.  High orderP-stable formulae for the numerical integration of periodic initial value problems , 1981 .

[4]  Guido Vanden Berghe,et al.  A modified numerov integration method for general second order initial value Problems , 1991, Int. J. Comput. Math..

[5]  E. Hairer.  Unconditionally stable methods for second order differential equations , 1979 .

[6]  M H Chawla,et al.  A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value , 1986 .

[7]  R. Thomas,et al.  Phase properties of high order, almostP-stable formulae , 1984 .

[8]  E. Hairer,et al.  A theory for Nyström methods , 1975 .

[9]  Ernst Hairer.  Méthodes de Nyström pour l'équation différentielley″=f(x, y) , 1976 .

[10]  H. De Meyer,et al.  A mono-implicit Runge-Kutta-Nyström modification of the Numerov method , 1997 .

[11]  J. M. Franco.  An explicit hybrid method of Numerov type for second-order periodic initial-value problems , 1995 .

[12]  J. Lambert,et al.  Symmetric Multistip Methods for Periodic Initial Value Problems , 1976 .

[13]  John P. Coleman,et al.  Numerical Methods for y″ =f(x, y) via Rational Approximations for the Cosine , 1989 .

[14]  H. De Meyer,et al.  A modified numerov integration method for second order periodic initial-value problems , 1990 .

[15]  M. Rizea,et al.  A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies , 1980 .

[16]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[17]  Guido Vanden Berghe,et al.  A general theory of stabilized extended one-step methods for odes , 1996, Int. J. Comput. Math..