Crank-Nicolson method

The above algorithm is leaves an ambiguity in the practical meaning of how to calculate the derivative evaluated in the central time-location of $t_{n+1/2} = (t_n + t_{n+1})/2$. To clarify this, we could regard this technique as a combination of a FTCS method, then a BTCS method, each with only half the total step-size in time. When the equations only have differential terms, this is straightforward, giving the obvious result that:

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

In one space dimension, this is called the Crank-Nicolson method, where it is easy to implement, since the resulting matrices are tri-diagonal.