ON OPEN BOUNDARIES IN THE FINITE ELEMENT APPROXIMATION OF TWO-DIMENSIONAL ADVECTION-DIFFUSION FLOWS

A steady-state and transient Þnite element model has been developed to approximate, with simple triangular elements, the two-dimensional advectionÐdi¤usion equation for practical river surface ßow simulations. Essentially, the spaceÐtime CrankÐNicolsonÐGalerkin formulation scheme was used to solve for a given conservative ßow-Þeld. Several kinds of point sources and boundary conditions, namely Cauchy and Open, were theoretically and numerically analysed. Steady-state and transient numerical tests investigated the accuracy of boundary conditions on inßow, noßow and outßow boundaries where di¤usion is important (di¤usive boundaries). With the proper choice of boundary conditions, the steady-state Galerkin and the transient CrankÐNicolsonÐGalerkin Þnite element schemes gave stable and precise results for advectiondominated transport problems. Comparisons indicated that the present approach can give equivalent or more precise results than other streamline upwind and high-order time-stepping schemes. Di¤usive boundaries can be treated with Cauchy conditions when the ßow enters the domain (inßow), and with Open conditions when the ßow leaves the domain (outßow), or when it is parallel to the boundary (noßow). Although systems with mainly di¤usive noßow boundaries may still be solved precisely with Open conditions, they are more susceptible to be inßuenced by other numerical sources of error. Moreover, the treatment of open boundaries greatly increases the possibilities of correctly modelling restricted domains of actual and numerical interest