A simple framework for the derivation and analysis of effective one-step methods for ODEs

作者：Brugnano Luigi; Iavernaro Felice^*; Trigiante Donato

来源：Applied Mathematics and Computation, 2012, 218(17): 8475-8485.

DOI：10.1016/j.amc.2012.01.074

• Ordinary differential equations
• Runge-Kutta methods
• One-step methods
• Hamiltonian problems
• Hamiltonian Boundary Value Methods
• Energy preserving methods
• Symplectic methods
• Energy drift

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 [2] and references therein). A few numerical tests are also included, in order to confirm the effectiveness of the methods resulting from our analysis.

• 出版日期2012-5-1