Abstract :
We solve a convection–diffusion–sorption (reaction) system on a bounded domain with
dominant convection using an operator splitting method. The model arises in contaminant
transport in groundwater induced by a dual-well, or in controlled laboratory experiments.
The operator splitting transforms the original problem to three subproblems: nonlinear
convection, nonlinear diffusion, and a reaction problem, each with its own boundary
conditions. The transport equation is solved by a Riemann solver, the diffusion one by
a finite volume method, and the reaction equation by an approximation of an integral
equation. This approach has proved to be very successful in solving the problem, but the
convergence properties where not fully known. We show how the boundary conditions
must be taken into account, and prove convergence in L1,loc of the fully discrete splitting
procedure to the very weak solution of the original system based on compactness
arguments via total variation estimates. Generally, this is the best convergence obtained
for this type of approximation. The derivation indicates limitations of the approach, being
able to consider only some types of boundary conditions. A sample numerical experiment
of a problem with an analytical solution is given, showing the stated efficiency of the
method
