CTCS Methods

To obtain higher accuracy and good stability, the semi-implicit or CTCS method is very useful, which combines both explicit and implicit schemes. We start by writing this method in a generic, abstract form:

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

From the Taylor expansion, the error due to the time-discretisation, for making each step in time is $O(\Delta t^3)$. Hence, the accumulated error in reaching a given time $t_N$ is approximately proportional to $\Delta t^2 = \Delta t^3/\Delta t$ in the CTCS methods. Thus, a much lower error can be achieved for the same time-step, while at the same time there is usually an increase in stability.