DuFort-Frankel method

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],... 

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). 

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. 

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]. 

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. 

In this first phase of the work the explicit finite difference method of Binder-Schmidt was applied W other methods being examined are the method of Dufort and Frankel (9) and orthogonal collocation (10). 

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