The Dufort-Frankel Method

We consider here one of many alternative algorithms which have been designed to overcome the stability problems of the simple algorithm. The Dufort-Frankel method is a trick which exploits the unconditional stability of the intrinsic method for simple differential equations. We modify (2.27) to read

$$u_j^{n+1} = u_j^{n-1} + {2\kappa\delta t\over\delta x^2} \... ...}^n - \left(u^{n+1}_j + u^{n-1}_j\right) + u_{j-1}^n\right\}.$$ (2.30)

which can be solved explicitly for $u_j^{n+1}$ at each mesh point

$$u^{n+1}_j = \left(1-\alpha\over 1+\alpha\right) u^{n-1}_j ... ...\alpha\over 1+\alpha\right)\left(u^n_{j+1} + u^n_{j-1}\right)$$ (2.31)

where

$$\alpha = 2{\kappa\delta t\over\delta x^2}.$$ (2.32)

When the usual von Neumann analysis is applied to this method it is found to be unconditionally stable. Note however that this does not imply that $\delta x$ and $\delta t$ can be made indefinitely large; common sense tells us that they must be small compared to any real physical time or length scales in the problem. We must still worry about the accuracy of the method. Another difficulty this method shares with the Leap-Frog method is that it requires boundary conditions at 2 times rather than one, even though the original diffusion equation is only 1st order in time.