Semi-implicit Euler method: Difference between revisions

Revision as of 22:23, 29 January 2014

Revision as of 12:46, 5 May 2014

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The stability region of the semi-implicit method was presented in <ref>[http://cat.inist.fr/?aModele=afficheN&cpsidt=1008675 Niiranen, J.: 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 method was misleadingly called symmetric Euler. 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 criteria is <math>s > - 2/\Delta t</math>

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

As can be seen, the semi-implicit method can simulate correctly both stable systems that have their roots in the left half plane and unstable systems that have their roots in the right half plane. This is clear advantage over forward (standard) Euler and backward Euler. Forward Euler tends to have less damping than the real system when the negative real parts of the roots get near the imaginary axis and backward Euler may show the system be stable even when the roots are in the right half plane.

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|isbn= 978-0-306-46631-1

|pages=page 117}}

{{Numerical integrators}}

[[Category:Numerical differential equations]]