Convergence of an operator splitting method on a bounded domain for a convection–diffusion–reaction system

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 L 1,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.

Contaminant Transport in Partially Saturated Porous Media

“…Using variation of constants, for we get (7) with , being the initial conditions. For we get (8) Computing the solution of ( 8) is very quick and as a result of the third step we set and . This scheme describes our numerical approximation of the original problem in the domain .…”

“…The most important feature of the proof is the determining of a sequence of a priori estimates of the solution , , in and the derivatives of the solution in . The estimates in are based on the maximum principle and can be generalized also for domain (see [8]). For the estimates of derivatives we follow a similar technique as in [4], where the convergence in was shown.…”