19-23 June 2023
Faculty of Physics, University of Warsaw
Europe/Warsaw timezone
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Generalization of the discrete gradient approach: improving accuracy and extending the scope of application

Presented by Jan CIEśLIńSKI on 22 Jun 2023 from 18:00 to 18:20


The motivation of our research is to find structure preserving discretizations of dynamical systems by developing some ideas related to the method of discrete gradients [1]. First, we recall that the discrete gradient method can be improved in two different ways without losing the energy conservation property, either by increasing its order [2], or by so-called locally exact discretizations that become extremely accurate in the vicinity of stable equilibria [3]. Second, we present our recent results concerning systems with e.g. non-linear damping or amplification (like Van der Pol oscillator or chaotic Rössler system). The approach based on the so-called reservoirs and para-Hamiltonian formulation, developed by the co-author [4], makes possible to extend the discrete gradient methods to a general class of autonomous ODEs (no integrals of motion are required). Side effect of this approach is a whole slew of new geometric methods. [1] R.I. McLachlan, G.R.W. Quispel, N. Robidoux: Geometric Integration using discrete gradients, Philosophical Transactions of the Royal Society A: Math. Phys. Eng. Sci. 357 (1998) 1021--1045. [2] J.L .Cieśliński, B. Ratkiewicz, Discrete gradient algorithms of high order for one-dimensional systems, Computer Physics Communications 183 (2012) 617--627. [3] J.L. Cieśliński: Locally exact modifications of discrete gradient schemes, Physics Letters A 377 (2013) 592-597. [4] A. Kobus, J.L. Cieśliński: Para-Hamiltonian form for General Autonomous ODE Systems: Introductory Results, Entropy 24 (3) (2022) 338.


Location: Faculty of Physics, University of Warsaw
Address: Pasteura 5, Warsaw
Room: Lecture hall: 0.06

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