Assessment of some numerical methods for estimating the parameters of the one-dimensional advection\u2013dispersion model

Abstract

This study appraised optimisations of numerical solutions of the one-dimensional advection–dispersion model (AD-Model) to synthetic data generated using an analytical solution. The motivation for the work was to identify reliable methods for estimating stream solute transport parameters from observed events in small rivers. Numerical solutions of the AD-Model must contend with several effects that might disturb the solution, with the introduction of numerical diffusion and numerical dispersion being particularly important issues. This poses a problem if physical dispersion is being identified by optimising model coefficients using observations of solute transport from field experiments. The discretisation schemes used were the Backward-Time/Centred-Space, Crank–Nicolson, Implicit QUICK, MacCormack and QUICKEST methods. Optimisations were obtained for several grid resolutions by keeping the time step constant whilst varying the space step: the range of Peclet number, Pe, was 1.5–12.0. Generally, increasing the space step led to poorer estimated coefficients and poorer fits to the synthetic concentration profiles. For Pe < 5 only Crank–Nicolson, MacCormack and QUICKEST gave reliable optimised dispersion coefficients: those from Backward-Time/Centred-Space and Implicit QUICK being significantly underestimated. For Pe > 5 Crank–Nicolson and MacCormack gave slightly overestimated dispersion coefficients whilst the other methods gave significantly underestimated dispersion coefficients. These findings were generally consistent with the known presence of numerical diffusion and numerical dispersion in the methods.