Semi-implicit Euler method: Difference between revisions

Revision as of 09:51, 29 September 2021

Revision as of 21:28, 16 October 2021

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Alternating between the two variants of the semi-implicit Euler method leads in one simplification to the Störmer-[[Verlet integration]] and in a slightly different simplification to the [[leapfrog integration]], increasing both the order of the error and the order of preservation of energy.<ref name="hairer2003" />

The stability region of the semi-implicit method was presented by Niiranen<ref>[https://www.researchgate.net/publication/268034494_Fast_and_accurate_symmetric_Euler_algorithm_for_electromechanical_simulations_NOTE_The_method_became_later_known_as_Symplectic_Euler Niiranen, Jouko: Fast and accurate symmetric Euler algorithm for electromechanical simulations] Proceedings of the Electrimacs'99, Sept. 14-16, 1999 Lisboa, Portugal, Vol. 1, pages 71 - 78.</ref> although the semi-implicit Euler was misleadingly called symmetric Euler in his paper. The semi-implicit method models the simulated system correctly if the complex roots of the characteristic equation are within the circle shown below. For real roots the stability region extends outside the circle for which the criteriacriterion is <math>s > - 2/\Delta t</math>

[[Image:Symplectic Euler stability region.jpeg]]