Consistency, Stability and Convergence of a Dufort–Frankel Scheme for a One-Dimensional Advection–Diffusion–Decay Model of Chlorine Transport in Contact Tanks

A three-level Dufort–Frankel scheme for the one-dimensional advection–diffusion–decay equation is stable for every diffusion number whenever the Courant number does not exceed 1, and is consistent only when the time step shrinks faster than the grid spacing. Chlorine concentration in a water-treatment contact tank is governed by advection, longitudinal dispersion and first-order decay, so a numerical model must be shown stable and consistent before it is used to predict disinfection performance. The decay term is evaluated at the two outer time levels, because a centred evaluation makes the leap-frog family unstable. Von Neumann analysis of the amplification polynomial, using the Schur–Cohn criterion, gives |g|≤1 for r ≥ 0 all and all decay numbers when 0 ≤ c= mk/h ≤ 1. A finite-grid eigenvalue check of the Toeplitz iteration matrix supports this. A Taylor-series analysis gives a truncation error of O(k^2,h^2,(k/h)^2), so the scheme is second-order accurate when is held fixed as the grid is refined. The Lax equivalence theorem then gives convergence under the same conditions. Numerical experiments against an exact solution confirm the theory: with fixed the observed order of accuracy reaches 2.00 for both r = 0.5 and r = 2, with the Courant number fixed the error stalls near 8×10^-2 however fine the grid, and on a periodic grid every run with c>1 grows while every run with c ≤ 1 does not.