THE DYNAMICS OF RUNGE–KUTTA METHODS

The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

[1]  P. Henrici.  Discrete Variable Methods in Ordinary Differential Equations , 1962 .

[2]  J. M. Sanz-Serna,et al.  Order conditions for canonical Runge-Kutta schemes , 1991 .

[3]  Robert L. Devaney Pallarés.  An introduction to chaotic dynamical systems , 1989 .

[4]  M.L. Liou,et al.  Computer-aided analysis of electronic circuits: Algorithms and computational techniques , 1977, Proceedings of the IEEE.

[5]  F. Lasagni.  Canonical Runge-Kutta methods , 1988 .

[6]  Celso Grebogi,et al.  Numerical orbits of chaotic processes represent true orbits , 1988 .

[7]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[8]  R. McLachlan,et al.  The accuracy of symplectic integrators , 1992 .

[9]  Curtis R. Menyuk,et al.  Some properties of the discrete Hamiltonian method , 1984 .

[10]  Feng Kang,et al.  Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study , 1991 .

[11]  Wolf-Jürgen Beyn,et al.  On invariant closed curves for one-step methods , 1987 .

[12]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[13]  H. C. Yee,et al.  Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I. The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics☆ , 1991 .

[14]  M. Bernhard.  Introduction to Chaotic Dynamical Systems , 1992 .

[15]  K. Feng.  Difference schemes for Hamiltonian formalism and symplectic geometry , 1986 .

[16]  Wojciech Rozmus,et al.  A symplectic integration algorithm for separable Hamiltonian functions , 1990 .

[17]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

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

[19]  E. A. Jackson,et al.  Perspectives of nonlinear dynamics , 1990 .

[20]  P. Kloeden,et al.  Stable attracting sets in dynamical systems and in their one-step discretizations , 1986 .

[21]  J. Marsden,et al.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators , 1988 .

[22]  James A. Yorke,et al.  Rigorous verification of trajectories for the computer simulation of dynamical systems , 1991 .

[23]  W. Beyn.  On the Numerical Approximation of Phase Portraits Near Stationary Points , 1987 .

[24]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[25]  David K. Arrowsmith,et al.  THE BOGDANOV MAP: BIFURCATIONS, MODE LOCKING, AND CHAOS IN A DISSIPATIVE SYSTEM , 1993 .

[26]  D. Aronson,et al.  Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study , 1982 .

[27]  Shigehiro Ushiki,et al.  Central difference scheme and chaos , 1982 .

[28]  H. Stetter.  Analysis of Discretization Methods for Ordinary Differential Equations , 1973 .

[29]  Richard H. Miller,et al.  A horror story about integration methods , 1991 .

[30]  Shigehiro Ushiki,et al.  Chaos in numerical analysis of ordinary differential equations , 1981 .

[31]  K. Tomita,et al.  10. Periodically forced nonlinear oscillators , 1986 .

[32]  T. Itoh,et al.  Hamiltonian-conserving discrete canonical equations based on variational difference quotients , 1988 .

[33]  Symmetry, Stability, Geometric Phases, and MechanicalIntegrators , 1991 .

[34]  A. Iserles.  Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations , 1990 .

[35]  Alex Friedman,et al.  Long-time behaviour of numerically computed orbits: small and intermediate timestep analysis of one-dimensional systems , 1991 .

[36]  M. Prüfer,et al.  Turbulence in multistep methods for initial value problems , 1985 .

[37]  C. Scovel,et al.  Symplectic integration of Hamiltonian systems , 1990 .

[38]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[39]  D. Earn,et al.  Exact numerical studies of Hamiltonian maps: iterating without roundoff error , 1992 .

[40]  L. Gardini,et al.  Bifurcations and transitions to chaos in the three-dimensional Lotka-Volterra map , 1987 .

[41]  R. Ruth,et al.  Fourth-order symplectic integration , 1990 .

[42]  J. Marsden,et al.  Symmetry, Stability, Geometric Phases, and Mechanical Integrators (Part II) , 1991 .

[43]  M. Kerimov,et al.  Modern numerical methods for ordinary differential equations , 1980 .

[44]  J. Sanz-Serna,et al.  Studies in numerical nonlinear instability III: Augmented Hamiltonian systems , 1987 .