The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: the case of symplecticity

DOI10.1016/j.amc.2011.03.022zbMath1245.65085arXiv1010.4538OpenAlexW2128757785MaRDI QIDQ440665

Donato Trigiante, Felice Iavernaro, Luigi Brugnano

Publication date: 19 August 2012

Published in: Applied Mathematics and Computation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1010.4538

zbMATH Keywords

symplectic methodsHamiltonian boundary value methodsenergy preserving methodshbvmsKutta collocation methodspolynomial HamiltonianRunge

Mathematics Subject Classification ID

Numerical methods for initial value problems involving ordinary differential equations (65L05)

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Cites Work

• Unnamed Item
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• Unnamed Item
• A simple framework for the derivation and analysis of effective one-step methods for ODEs
• State-dependent symplecticity and area preserving numerical methods
• Reprint of ``Analysis of Hamiltonian boundary value methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
• A family of trigonometrically fitted partitioned Runge-Kutta symplectic methods
• Symplectic methods for the numerical solution of the radial Schrödinger equation
• Preserving algebraic invariants with Runge-Kutta methods
• An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants
• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
• Simulating Hamiltonian Dynamics
• Geometric integration using discrete gradients
• Energy-preserving Runge-Kutta methods
• A new class of energy-preserving numerical integration methods
• Geometric Numerical Integration
• s-stage Trapezoidal Methods for the Conservation of Hamiltonian Functions of Polynomial Type
• Hamiltonian BVMs (HBVMs): A Family of “Drift Free” Methods for Integrating polynomial Hamiltonian problems

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