Discrete gradient algorithms of high order for one-dimensional systems

2010/08/23 by Jan L. Cieśliński, Cieśliński, Jan L., Bogusław Ratkiewicz +1
Computer Science · Mathematics · Physics and Astronomy · #Computational Physics (physics.comp-ph) #FOS: Mathematics #FOS: Physical sciences #Numerical Analysis (math.NA) #cs.NA #math.NA #physics.comp-ph

paper · pdf · doi:10.48550/arxiv.1008.3895

22 pages, 9 figures

arxiv created 2010/08/23 · arxiv updated 2010/08/24

Abstract

We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories in the phase space. The accuracy of our integrators is higher by several orders of magnitude as compared with the standard discrete gradient scheme (modified midpoint rule) and, what is more, our schemes have very high accuracy even for large time steps.

Related

• Order theory for discrete gradient methods
• Higher order discrete gradient method by the discontinuous Galerkin time-stepping method
• Computational enhancement of discrete gradient method
• Locally exact modifications of numerical integrators
• Discrete gradient methods for preserving a first integral of an ordinary\n differential equation