Multi-Point Second Spatial Derivatives

In practice, the (6,5) approach is, at present, limited by the fact that, in the form presented here, it applies to equal intervals. A slight improvement with unequal intervals, using a 4-point spatial second derivative, is described in Chap. 8, and this might be sufficient improvement, at little cost in terms of desk work [143]. It has been applied to the ultramicroelectrode [532], see Chap. 12. 

The (ode-) method called leapfrog has been mentioned in Chap. 4, where (4.38) describes it. This was used by Richardson [468] to solve a parabolic pde, apparently with success. The computational molecule corresponding to this method is 

In this scheme, the temporal derivative is formed by the central (second-order ) difference between the upper and lower points, the second spatial derivative being approximated as usual. This makes the discretisation at the index i in space, 

Leapfrog is used with apparent success to solve hyperbolic pdes [528], but was proved unconditionally unstable for parabolic pdes in 1950 [424]. Richardson had been lucky, in that the instabilities had not made themselves felt in his (pencil and paper) calculations, in the course of the few iterations he worked. 

DuFort and Frankel [216] devised a modification to this scheme in 1953 that stabilises it  

---

Britz D (2003) Higher-order spatial discretisations in digital simulations. Algorithm for any multi-point first- or second derivative on an arbitrarily spaced grid. Electrochem Commun 5 195-198... 

---