REVISITING FINITE DIFFERENCE SOLUTIONS FOR HEAT-TYPE EQUATIONS. PART III. DIFFUSION-ADVECTION TRANSFER

In this third paper of a series, we revisit all major sysematic uncertainties that affect a complete and unbiased sample of twelve finite difference schemes for diffusion-advectionlike equations. In order to provide the coherent picture,unlike the existing way, we use as the key tenets both thereverse Taylor’s analysis and the discrete Fourier’s analysis, as well as the monotonicity analysis. For every typeof scheme, their theoretical uncertainties are examined. Adetailed graphical investigation is also provided. We find that no scheme considered in this study resolves the smallerlength scales well. We report numerous unexpected results,almost all of them related to the stability issue. Furthermore, we present several numerical experiments on an equalfooting corroborating our demonstrations and proving towhat degree the accuracy of each scheme is impaired by thediscontinuities in the data. A comparison with each otheris made as well. We definitively establish that the CrankNicolson scheme is preferred by experiments.Math. Subject Classification:35Q68, 35K05, 35K10,65M06, 65N22, 97N40