Numerical modelling of solute transport processes using higher order accurate finite difference schemes. Numerical treatment of flooding and drying in tidal flow simulations and higher order accurate finite difference modelling of the advection diffusion equation for solute transport predictions.

摘要

The modelling of the processes of advection and dispersion-diffusion is theudmost crucial factor in solute transport simulations. It is generally appreciatedudthat the first order upwind difference scheme gives rise to excessive numericaluddiffusion, whereas the conventional second order central difference scheme exhibitsudsevere oscillations for advection dominated transport, especially in regionsudof high solute gradients or discontinuities. Higher order schemes have thereforeudbecome increasingly used for improved accuracy and for reducing grid scale oscillations.udTwo such schemes are the QUICK (Quadratic Upwind Interpolation forudConvective Kinematics) and TOASOD (Third Order Advection Second OrderudDiffusion) schemes, which are similar in formulation but different in accuracy,udwith the two schemes being second and third order accurate in space respectivelyudfor finite difference models. These two schemes can be written in variousudfinite difference forms for transient solute transport models, with the differentudrepresentations having different numerical properties and computational efficiencies.udAlthough these two schemes are advectively (or convectively) stable,udit has been shown that the originally proposed explicit QUICK and TOASODudschemes become numerically unstable for the case of pure advection. The stabilityudconstraints have been established for each scheme representation basedudupon the von Neumann stability analysis. All the derived schemes have beenudtested for various initial solute distributions and for a number of continuousuddischarge cases, with both constant and time varying velocity fields.udThe 1-D QUICKEST (QUICK with Estimated Streaming Term) scheme isudthird order accurate both in time and space. It has been shown analytically andudnumerically that a previously derived quasi 2-D explicit QUICKEST scheme,udwith a reduced accuracy in time, is unstable for the case of pure advection. Theudmodified 2-D explicit QUICKEST, ADI-TOASOD and ADI-QUICK schemesudhave been developed herein and proved to be numerically stable, with the bility sta- region of each derived 2-D scheme having also been established. All theseudderived 2-D schemesh ave been tested in a 2-D domain for various initial solute distributions with both uniform and rotational flow fields. They were furtherudtested for a number of 2-D continuous discharge cases, with the correspondingudexact solutions having also been derived herein.udAll the numerical tests in both the 1-D and 2-D cases were compared withudthe corresponding exact solutions and the results obtained using various otheruddifference schemes, with the higher order schemes generally producing moreudaccurate predictions, except for the characteristic based schemes which failedudto conserve mass for the 2-D rotational flow tests. The ADI-TOASOD schemeudhas also been applied to two water quality studies in the U. K., simulating nitrateudand faecal coliform distributions respectively, with the results showing audmarked improvement in comparison with the results obtained by the secondudorder central difference scheme.udDetails are also given of a refined numerical representation of flooding anduddrying of tidal flood plains for hydrodynamic modelling, with the results showingudconsiderable improvements in comparison with a number of existing modelsudand in good agreement with the field measured data in a natural harbour study.