On Open Boundaries in the Finite Element Approximation of Two-dimensional Advection-diffusion Flows

A steady-state and transient finite element model has been developed to approximate, with simple triangular elements, the two-dimensional advection—diffusion equation for practical river surface flow simulations. Essentially, the space—time Crank—Nicolson—Galerkin formulation scheme was used to solve for a given conservative flow-field. 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 inflow, noflow and outflow boundaries where diffusion is important (diffusive boundaries). With the proper choice of boundary conditions, the steady-state Galerkin and the transient Crank—Nicolson—Galerkin finite 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. Diffusive boundaries can be treated with Cauchy conditions when the flow enters the domain (inflow), and with Open conditions when the flow leaves the domain (outflow), or when it is parallel to the boundary (noflow). Although systems with mainly diffusive noflow boundaries may still be solved precisely with Open conditions, they are more susceptible to be influenced 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. ( 1997 by John Wiley & Sons, Ltd.