A two-step, fourth-order method with energy preserving properties
- 作者：Luigi Brugnano<sup>asup><sup> ; sup><sup>luigi.brugnano@unifi.itsup> ; Felice Iavernaro<sup>bsup><sup> ; sup><sup>felix@dm.uniba.itsup> ; Donato Trigiante<sup>1sup>
- 关键词：Ordinary differential equations ; Mono-implicit methods ; Multistep methods ; One-leg methods ; Canonical Hamiltonian problems ; Hamiltonian boundary value methods ; Energy preserving methods ; Energy drift
- 刊名：Computer Physics Communications
- 出版年：2012
- 期刊代码：30_00104655
- 类别：cp
- 出版时间：September, 2012
- 卷：183
- 期：9
- 页码：1860-1868
- 文件大小：1283 K
摘要
We introduce a family of fourth-order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. As is the case with linear mutistep and one-leg methods, a prerogative of the new formulae is that the associated nonlinear systems to be solved at each step of the integration procedure have the very same dimension of the underlying continuous problem.
The key tools in the new methods 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.