Time discretisation errors

We have assumed (to a first approximation) that the time derivative of the field $\mbf{A}$ doesn't vary in the small time interval from $t_n$ to $t_{n+1}$. This is obviously a better approximation when $\Delta t$ is small. However, it is useful to think about this more quantitatively. From the Taylor expansion, the error for making each step in time is $O(\Delta t^2)$. However, one must take $N$ steps of size $\Delta t$ to reach a given time $t_N$ , where $N = (t_N - t_0)/\Delta t$ . Hence, the accumulated error in reaching a given time $t_{N}$ is approximately proportional to $\Delta t = \Delta t^2 / \Delta t$ in the FTCS methods, provided the error simply adds, and does not grow exponentially. This error can therefore be reduced by halving the time-step, until the changes that occur are less than the error tolerance.