A simple framework for the derivation and analysis of effective one-step methods for ODEs
- 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>csup><sup> ; sup><sup>1sup>
- 关键词：Ordinary differential equations ; Runge&ndash ; Kutta methods ; One-step methods ; Hamiltonian problems ; Hamiltonian Boundary Value Methods ; Energy preserving methods ; Symplectic methods ; Energy drift
- 刊名：Applied Mathematics and Computation
- 出版年：2012
- 期刊代码：7_00963003
- 类别：cp
- 出版时间：1 May, 2012
- 卷：218
- 期：17
- 页码：8475-8485
- 文件大小：547 K
摘要
In this paper, we provide a simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods. The framework consists in coupling two simple tools: firstly a local Fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge-Kutta methods can be derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see and references therein). A few numerical tests are also included, in order to confirm the effectiveness of the methods resulting from our analysis.