A Two Step, Fourth Order, Nearly-Linear Method with Energy Preserving Properties
arXiv:1106.0598 · doi:10.1016/j.cpc.2012.04.002
Abstract
We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction term is O(h^5) (h is the stepsize of integration). The key tools the new methods are based upon are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results.
14 pages, 4 figures, 2 tables
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- Line Integral solution of Hamiltonian PDEs
- A note on the continuous-stage Runge-Kutta-(Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)
- Line Integral Solution of Hamiltonian Systems with Holonomic Constraints
- An explicit and practically invariants-preserving method for conservative systems
- A novel class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation
- LINE INTEGRAL METHODS and their application to the numerical solution of conservative problems
- Continuous-Stage Runge-Kutta approximation to Differential Problems