5.2.4 Dufort-Frankel Scheme (DF)

DST in time and space, but in place of $-2u_{j}^{n}$ use $-(u_{j}^{n+1}+u_{j}^{n-1})$:

$$\frac{1}{2 \Delta t} \left[ u_{j}^{n+1}-u_{j}^{n-1} \right] = \frac{\lambda}{(\Delta x)^{2}} \left[ u_{j+1}^{n}-(u_{j}^{n+1}+u_{j}^{n-1})+u_{j-1}^{n} \, \right]$$

or, with $a \equiv 2 \lambda \Delta t/(\Delta x)^{2}$,

$$\fbox{$ \displaystyle u_{j}^{n+1} = \frac{1-a}{1+a}u_{j}^{n-1} + \frac{a}{1+a} \left[ u_{j+1}^{n}+ u_{j-1}^{n}\right] $}$$

The DF algorithm is of second order in $\Delta t$. In contrast to CN, it is an explicit expression for $u_{j}^{n+1}$.

Stability:

$$g = \frac{1}{1+a} \left[ a \, \cos \, k \Delta x \pm \sqrt{1-a^{2}\sin^{2}k \Delta x} \, \right]$$

Considering in turn the cases $a^{2}\sin^{2}k \Delta x \geq 1$ and $\dots < 1$ we find that $\vert g\vert^{2}\leq 1$ always; the method is unconditionally stable.