$$ \newcommand{\D}{\displaystyle} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})} $$




7.5.2 Dufort-Frankel scheme (1953)

Richardson scheme in (7.57) can be made stable by the following modification: $$ \begin{equation} \tag{7.58} u^n_j=\frac{1}{2}(u_j^{n+1}+u^{n-1}_j) \end{equation} $$

which inserted in (7.57) gives: $$ \begin{equation} \tag{7.59} u^{n+1}_j=\frac{1}{1+2D}\left[ (1-2D)u^{n-1}_j+2D(u^n_{j+1}+u^n_{j-1})\right] \end{equation} $$

This is an explicit 3-level scheme called DuFort-Frankel scheme, see Figure 93.

Figure 93: 3-level stencil of the DuFort-Frankel scheme (7.59).

3-level schemes where \( u^n_j \) is missing are called Leap-frog-type schemes for obvious reasons. The sufficient criterion of positive coefficients in (7.21) requires that \( D\leq \frac{1}{2} \) for a stable scheme. The stability analysis here is slightly more complicated as with previous cases because we have to discuss what happens when \( G \) takes complex values.

Inserting (7.42) in (7.59) and division by \( G^{n-1}e^{e^{i\delta j}} \) yields: $$ \begin{equation*} G^2= \frac{1}{1+2r}\left[(1-2r)+2r\cdot G\cdot (e^{i\delta}+e^{-i\delta})\right] = \frac{1}{1+2r}\left[(1-2r)+4r\cdot G\cdot \cos(\delta)\right] \end{equation*} $$ which gives the following 2nd order equation: $$ \begin{equation*} (1+2D)\cdot G^2-4D\cdot G\cos(\delta)-(1-2D)=0\,, \end{equation*} $$ with roots: $$ \begin{equation*} \begin{array}{lcl} G_{1,2} & = & \dfrac{4D\cos(\delta) \pm \sqrt{(4D\cos(\delta))^2 +4(1+2D)\cdot(1-2D)}}{2(1+2D)}\\ & = & \dfrac{2D\cos(\delta)\pm \sqrt{1-D^2\sin^2(\delta)}}{1+2D}. \end{array} \end{equation*} $$

For stability, both roots must meet the condition \( |G|\leq 1 \). In general, we must distinguish between real and complex roots to take care of the case for which \( G\leq 0 \) when \( G \) is real.

  1. Real roots: \( 1-4D^2\sin(\delta)\geq0 \)
    \( |G_{1,2}| \leq \dfrac{2D\cdot |\cos(\delta)|+\sqrt{1-4D^2\sin^2(\delta)}}{1+2r}\leq \dfrac{1+2D}{1+2D}\leq1 \)

  2. Complex roots: \( 1-4D^2\sin^2(\delta) < 0 \to \sqrt{1-4D^2\sin^2(\delta)}=i\cdot \sqrt{4D^2\sin^2(\delta)-1} \)

    \( |G_{1,2}|^2= \bigg|\frac{\big(2D\cos(\delta)\big)^2+4D^2\sin^2(\delta)-1}{(1+2D)^2}\bigg|=\bigg|\frac{4D^2-1}{4D^2+4D+1}\bigg|=\bigg|\frac{2D-1}{2D+1}\bigg| < 1 \)
Analysis shows that (7.59) is actually unconditionally stable. The DuFort-Frankel scheme is the only simple know explicit scheme with 2nd order accuracy in space and time that has this property. Therefore it has been in part used to solve the Navier-Stokes equations. In section (7.6 Truncation error, consistency and convergence) we shall see that there is however a severe problem with this scheme. For the first time level one lacks the lowest time level required by the scheme. One option is to evolve the solution for the first time step using the FTCS scheme.