Semi-implicit Euler method

The semi-implicit Euler method produces an approximate discrete solution by iterating

v_{n+1} = v_n + g(t_n, x_n) \, \Delta t \quad

x_{n+1} = x_n + f(t_n, v_{n+1}) \, \Delta t \quad

where Δt is the time step and t_n = t_0 + n\,\Delta t is the time after n steps.

The difference with the standard Euler method is that the semi-implicit Euler method uses vn + 1 in the equation for xn + 1, while the Euler method uses vn.

The semi-implicit Euler is a first-order integrator, just as the standard Euler method. This means that it commits a global error of the order of Δt. 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.


From wikipedia: http://en.wikipedia.org/wiki/Symplectic_Euler_method