Semi-implicit Euler method: Difference between revisions

Revision as of 17:00, 17 June 2008

Revision as of 12:50, 13 April 2009

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:<math> x_{n+1} = x_n + f(t_n, v_{n+1}) \, \Delta t \quad</math>

where <math> \Delta Δ''t </math>'' is the time step and ''t<mathsub>t_nn</sub>'' = t_0''t₀'' + ''n\,\Delta ''Δ''t</math>'' is the time after ''n'' steps.

The difference with the standard Euler method is that the semi-implicit Euler method uses ''v''<mathsub>v_{''n''+1}</mathsub> in the equation for ''v''<mathsub>x_{''n''+1}</mathsub>, while the Euler method uses ''v<mathsub>v_nn</mathsub>''.

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.