A Fully Conservative Eulerian-Lagrangian Stream-Tube Method for Advection-Diffusion Problems

We present a new method for a two-dimensional linear advection-diffusion problem of a “tracer” within an ambient fluid. The problem should have isolated external sources and sinks, and the bulk fluid flow is assumed to be governed by an elliptic problem approximated by a standard locally conservative scheme. The new method, the fully conservative Eulerian-Lagrangian streamtube method, combines the volume corrected characteristics-mixed method with the use of a streamtube mesh. Advection of the tracer is approximated using characteristic tracing in time of regions of space, which maintains mass conservation. However, the shape of a characteristic trace-back region is numerically approximated, so its volume must also be correct to maintain accurate approximation of the tracer density (i.e., the mass of the ambient fluid must be conserved during the advection step). Our new method has the advantages that it is fully locally conservative (both tracer and ambient fluid mass is conserved locally), has low numerical diffusion overall and no numerical crossdiffusion between stream-tubes, can use very large time steps (perhaps 20 to 30 times the CFL limited step), and can use a very coarse mesh, since it is tailored to the flow pattern. Because advection is approximated within stream-tubes, it is essentially one-dimensional, making it relatively easy to implement and computationally efficient. We also present a grid transfer technique to approximate more simply the physical diffusion on a rectangular grid rather than on the stream-tube mesh. The new method can be used for many applications, but especially problems of flow and transport in porous media, which have sources and sinks isolated to wells. Examples include the modeling of groundwater contaminant migration, petroleum production, and carbon sequestration.