DuFort-Frankel

The ode-) method called leapfrog has been mentioned in Chap. 4, where Eq. (4.38) describes it. This was used by Richardson [17] 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 [32], but was proved unconditionally unstable for parabolic pdes in 1950 [18]. 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 [33] devised a modification to this scheme in 1953 that stabilises it  

The time derivative is the same central difference but the spatial second derivative now leaves out the central point, substituting for it the mean of the past and future points. Thus, the discretisation is 

---

For convenience, the two species concentrations are given the symbols A and B here, with A and B the unknowns. Lerke et al. mention this briefly [361], for the DuFort-Frankel method [2161, for some time a method suggested bv Feldberg [233],... 

The DuFort-Frankel scheme has apparently been dropped in favour of more interesting schemes such as BDF, which can be driven to higher orders, and for which the start-up problem has been overcome (Chap. 4). 

Both the LR and RL variants, despite being explicit, are said to be stable for all A values, which is a great advantage. Also, the method does not share with DuFort-Frankel and hopscotch the propagational inadequacy problem [232] mentioned above because both variants amount to a recursive algorithm, each newly calculated element carrying with it some component from all previously calculated elements. 

Most discrete approximations that have been mentioned in this book are consistent, with the exception of one. This is the DuFort-Frankel method [216], described on page 153 in Chap. 9. It is stable for all A, yet it has a consistency problem. Giving (9.19) the same treatment as above, one ends with... 

This sort of analysis can be applied to other methods. Britz and Strutwolf [152] applied it to the BDF method using 5-point discretisation along X, and, also for 5-point approximations, Strutwolf and Britz [531] applied it to extrapolation. For a multilevel method such as BDF, the analysis results in a polynomial in , and complex roots are possible. For example, Lapidus and Pinder [350] treat the DuFort-Frankel method it results in a quadratic equation in but it is clear that is is unconditionally stable (even though we have seen that is not consistent). 

DuFort-Frankel Chap. 9, Sect. 9.2.3. Stable and explicit but inconsistent and shares the propagational inadequacy problem. 

A variety of explicit (Dufort-Frankel, Lax-Wendroff, Runge-Kutta) and implicit (approximate factorization, LU-SGS) or hybrid schemes have been employed for integration in time. Because of the complexity of the incompressible Navier-Stokes equations, stability analyses to determine critical time steps are difficult. As a general rule, the allowable time step for an explicit method is proportional to the ratio of the smallest grid size to the largest convective velocity (or the wave propagation speed for an artificial compressibility method). 

Several different integration methods are available in the literature. Many of them approximate the effect of the Laplace operator by finite differences [32,39]. Among the more elaborate schemes of this kind we mention the Crank-Nicolson scheme [32] and the Dufort-Frankel scheme [39]. 

In the class of implicit schemes there is a scheme of the second order accuracy in time, where the temperature values are taken as the half-sum of the values from the upper and lower layers. Such a scheme is not monotonous and wasn t used during the software package creation. For the same reason an explicit three-layer Dufort-Frankel scheme (rhombus) was rejected. In this case the time difference is calculated with values through two layers and the temperature flows in the central point are calculated as a half-sum of the values on the lower and the upper layer [2, 7], This scheme is absolutely stable but requires a small time step for aceuracy achievement. 

The discretization of the temperature equation is based on a finite-difference method (FDM), where different solution procedures have been used for steady and unsteady flows namely the Jacobi algorithm for steady state flows and the DuFort-Frankel approach for transient flows [20]. For the numerical calculation of the temperature field by the FDM, the same numerical grid was taken as for the LBM. Such a regular grid is very much suitable for finite-difference methods. 

---