Semi-implicit Method

    Another concern in the numerical simulation is the time variable in Eq. (19).  The explicit forward Euler method requires very small time step to maintain stability.  Instead, we adopt the semi-implicit scheme, which has been used by other scholars in the simulation of different systems.  The key idea is treat the linear term implicitly to reduce the stability constraint, while still treat the nonlinear term explicitly to avoid solving nonlinear equations at each time step.

    For a given time t and a time step, , denote , , .  In eq.  (22) replace  by  ,  by  , and  by .  We obtain that

This equation is valid for any  and .