Symplectic exponentially-fitted four-stage Runge–Kutta methods of the Gauss type

The construction of symmetric and symplectic exponentially-fitted Runge–Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions deserves a lot of interest. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new such integrators of eighth-order are studied and constructed by making use of the six-step procedure of Ixaru and Vanden Berghe (2004). Numerical experiments for some oscillatory problems are presented.

[1]  G. Berghe,et al.  Symplectic Exponentially-Fitted Modified Runge-Kutta Methods of the Gauss Type: Revisited , 2011 .

[2]  Theodore E. Simos,et al.  A Modified Runge-Kutta Method with Phase-lag of Order Infinity for the Numerical Solution of the Schrödinger Equation and Related Problems , 2001, Comput. Chem..

[3]  S. Yau.  Mathematics and its applications , 2002 .

[4]  G. Vanden Berghe,et al.  Exponential fitted Runge--Kutta methods of collocation type: fixed or variable knot points? , 2003 .

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

[6]  T. E. Simos,et al.  An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions , 1998 .

[7]  Higinio Ramos,et al.  Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations , 2003 .

[8]  H. De Meyer,et al.  Exponentially-fitted explicit Runge–Kutta methods , 1999 .

[9]  Manuel Calvo,et al.  Sixth-order symmetric and symplectic exponentially fitted modified Runge-Kutta methods of Gauss type , 2008, Comput. Phys. Commun..

[10]  T. E. Simos,et al.  Exponentially fitted symplectic integrator. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  D. Hollevoet,et al.  Three‐Stage Two‐Parameter Symplectic, Symmetric Exponentially‐Fitted Runge‐Kutta Methods of Gauss Type , 2010 .

[12]  H. De Meyer,et al.  Exponentially fitted Runge-Kutta methods , 2000 .

[13]  Jesús Vigo-Aguiar,et al.  AN ADAPTED SYMPLECTIC INTEGRATOR FOR HAMILTONIAN PROBLEMS , 2001 .

[14]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[15]  K. Ozawa.  A functional fitting Runge-Kutta-Nyström method with variable coefficients , 2002 .

[16]  D. G. Bettis.  Runge-Kutta algorithms for oscillatory problems , 1979 .

[17]  Manuel Calvo,et al.  Structure preservation of exponentially fitted Runge-Kutta methods , 2008 .

[18]  T. E. Simos,et al.  Controlling the error growth in long–term numerical integration of perturbed oscillations in one or several frequencies , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  Kazufumi Ozawa,et al.  A functional fitting Runge-Kutta method with variable coefficients , 2001 .

[20]  G. J. Cooper.  Stability of Runge-Kutta Methods for Trajectory Problems , 1987 .

[21]  Hans Van de Vyver.  A fourth-order symplectic exponentially fitted integrator , 2006, Comput. Phys. Commun..

[22]  Manuel Calvo,et al.  Sixth-order symmetric and symplectic exponentially fitted Runge-Kutta methods of the Gauss type , 2009 .

[23]  J. M. Franco.  Runge-Kutta methods adapted to the numerical integration of oscillatory problems , 2004 .

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

[25]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[26]  J. M. Sanz-Serna,et al.  Symplectic integrators for Hamiltonian problems: an overview , 1992, Acta Numerica.

[27]  G. Berghe,et al.  Fourth-order symplectic exponentially-fitted modified Runge-Kutta methods of the Gauss type: a review , 2010 .