Bootstrapping discrete-gradient integral-preserving integrators to fourth order

Ordinary differential equations having a first integral may be solved numerically using one of several methods, with the integral preserved to machine accuracy. One such method is the discrete gradient method. It is shown here that the order of the discrete gradient method can be bootstrapped repeatedly to higher orders of accuracy. The potential for improved efficiency offered by the bootstrapped method is illustrated using three 6-dimensional systems.

[1]  Robert I. McLachlan,et al.  Lie group foliations: dynamical systems and integrators , 2003, Future Gener. Comput. Syst..

[2]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

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

[4]  Robert I. McLachlan,et al.  On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods , 1995, SIAM J. Sci. Comput..

[5]  G. S. Turner,et al.  Discrete gradient methods for solving ODEs numerically while preserving a first integral , 1996 .

[6]  B. Leimkuhler,et al.  Simulating Hamiltonian Dynamics , 2005 .

[7]  G. Quispel,et al.  Integral-preserving integrators , 2004 .

[8]  L. Shampine.  Conservation laws and the numerical solution of ODEs, II , 1999 .

[9]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[10]  J. C. Simo,et al.  On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry , 1996 .

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

[12]  Arieh Iserles,et al.  Geometric integration: numerical solution of differential equations on manifolds , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[13]  G. Quispel,et al.  Geometric integrators for ODEs , 2006 .

[14]  E. Hairer.  Symmetric Projection Methods for Differential Equations on Manifolds , 2000 .

[15]  M. P. Laburta.  Construction of starting algorithms for the RK-Gauss methods , 1998 .

[16]  G. Quispel,et al.  A new class of energy-preserving numerical integration methods , 2008 .

[17]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[18]  G. Quispel,et al.  Foundations of Computational Mathematics: Six lectures on the geometric integration of ODEs , 2001 .

[19]  Qin Meng-Zhao,et al.  Construction of higher order symplectic schemes by composition , 2006, Computing.

[20]  G. Quispel,et al.  Geometric integration using discrete gradients , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[21]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[22]  M. Suzuki,et al.  Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations , 1990 .

[23]  G. Quispel,et al.  Solving ODEs numerically while preserving a first integral , 1996 .

[24]  G. Quispel,et al.  Acta Numerica 2002: Splitting methods , 2002 .

[25]  O. Gonzalez.  Time integration and discrete Hamiltonian systems , 1996 .

[26]  G.R.W. Wuispel,et al.  Volume-preserving integrators , 1995 .

[27]  R. McLachlan,et al.  Explicit Lie-Poisson integration and the Euler equations. , 1993, Physical review letters.

[28]  Thomas P. Weissert,et al.  The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem , 1999 .

[29]  C. W. Gear,et al.  Maintianing solution invariants in the numerical solution of ODEs , 1986 .

[30]  Uri M. Ascher,et al.  On Some Difficulties in Integrating Highly Oscillatory Hamiltonian Systems , 1999, Computational Molecular Dynamics.

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