An ELLAM Scheme for Advection-Di usion Equations in Multi-Dimensions

Advection-diiusion equations describe the transport of solutes in groundwater and surface water, the movement of aerosols and trace gases in the atmosphere, and many problems in other important applications. They often cause serious numerical diiculties. Classical space-centered or upwind numerical methods tend to yield numerical solutions with various combinations of excessive oscillations and numerical dispersion. In this paper we develop an Eulerian-Lagrangian localized adjoint method (ELLAM) for the solution of multi-dimensional advection-diiusion equations in a mass-conservative manner. ELLAM was originally developed by Celia, Russell, Herrera and Ewing, based on the general framework of Her-rera. Although computational advantages of ELLAM approximations have been demonstrated for a variety of one-dimensional transport equations, practical implementation of ELLAM schemes in multiple spatial dimensions requires careful algorithm development. The ELLAM scheme in this paper treats the advective part by a characteristic algorithm and approximates the diiusive part on a static Eulerian grid. A forward tracking algorithm is employed to evaluate some non-conventional terms on the right-hand side of the scheme to guarantee mass conservation and accuracy, which is a delicate issue in the implementation of characteristic methods. The scheme developed in this paper is tested for the transport of a Gaus-sian disturbance in a two-dimensional rotating ow eld. The numerical solutions obtained are compared with the analytical solution, and with the numerical solutions obtained by the Galerkin nite element method (GAL) and by the quadratic Petrov-Galerkin nite element method (QPG). With the same spatial grid, the ELLAM scheme generates much more accurate numerical solutions than GAL and QPG, even though they use a much ner time step than that for ELLAM. Moreover, the ELLAM solution is free of undershoot/overshoot and maintains the correct physical behavior of the analytical solution. In contrast, the GAL and QPG schemes produce numerical solutions with excessive oscillation, dispersion, deformation and phase error.