Semi-implicit Euler method: Difference between revisions

Revision as of 05:56, 31 May 2008

Revision as of 05:57, 31 May 2008

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where <math> \Delta t </math> is the time step and <math>t_n = t_0 + n\Delta t</math> is the time after ''n'' steps.

The difference with the standard Euler method is that the Euler–Cromersemi–implicit Euler method uses <math>v_{n+1}</math> in the equation for <math>x_{n+1}</math>, while the Euler method uses <math>v_n</math>.

The semi–implicit Euler is a [[Numerical ordinary differential equations#Consistency and order|first-order integrator]], just as the standard Euler method. This means that it commits a global error of the order of &Delta;t. However, the semi–implicit Euler method is a [[symplectic integrator]], unlike the standard method. As a consequence, the semi–implicit Euler method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.