Split-step method

The pure BCTS method is somewhat inefficient to implement when there are nonlinear terms present, as the solution of the nonlinear implicit equations couples all the different spatial points together. For this reason, it is more efficient to implement this in two stages, with an approach called a split-step method. This method still has a local discretisation error proportional to $O(\Delta t^3)$, and can be written in greater detail as:

$$\mbf{A}(t_{n+1}) = \mbf{A}(t_n) + \Delta t [\mbf{D}(\mbf{A}(t_n) + \mbf{A}(t_{n+1}))/2 + \mbf{U}[\mbf{A}(t_{n+1/2})] + O(\Delta t^3),$$ (B.3)

In this last equation, there is an implied summation over all the vector indices (if they exist) and lattice indices: Thus, the total method can be written in two steps as:

$$\mbf{A}(t_{n+1/2}) = [1+\Delta t \mbf{D}/2]\mbf{A}(t_{n})+(\Delta t/2)\mbf{U}[\mbf{A}(t_{n+1/2})]$$ (B.4)

$$\mbf{A}(t_{n+1}) = [1 - \Delta t \mbf{D}/2]^{-1} (\mbf{A}(t_{n+1/2}) + (\Delta t/2) \mbf{U}[\mbf{A}(t_{n+1/2})]),$$ (B.5)

It should be noted that the term $\mbf{U}[\mbf{A}(t_{n+1/2})]$ must be calculated implicitly--this can be achieved iteratively, as in the BTCS method treated previously.