Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. SCI. COMPUT. c  2012 Society for Industrial and Applied Mathematics Vol. 34, No. 4, pp. B447–B478 A FULLY CONSERVATIVE EULERIAN-LAGRANGIAN STREAM-TUBE METHOD FOR ADVECTION-DIFFUSION PROBLEMS ∗ TODD ARBOGAST † , CHIEH-SEN HUANG ‡ , AND CHEN-HUI HUNG § Abstract. 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 stream- tube method, combines the volume corrected characteristics-mixed method with the use of a stream- tube 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 cross- diffusion 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. Key words. Eulerian-Lagrangian, stream-tube, finite volume, locally conservative, characteris- tics, hyperbolic transport, porous media, grid transfer AMS subject classifications. 65M08, 65M25, 65M60, 76M12, 76R50, 76S05 DOI. 10.1137/110840376 1. Introduction. We consider the linear hyperbolic transport problem with iso- lated external sources and sinks in which one component (say, a tracer) is predomi- nately advected but also mildly diffused within an ambient fluid. For such problems, characteristic or Eulerian-Lagrangian (or semi-Lagrangian) methods have the advan- tages that long time steps can be used without loss of stability, numerical diffusion can be low, and relatively coarse computational meshes can be used effectively. An important application is to flow in a porous medium with isolated wells, modeling, e.g., groundwater contaminant migration, petroleum production, and carbon seques- ∗ Submitted to the journal’s Computational Methods in Science and Engineering section July 11, 2011; accepted for publication (in revised form) May 14, 2012; published electronically August 9, 2012. http://www.siam.org/journals/sisc/34-4/84037.html † Department of Mathematics, University of Texas at Austin, Austin, TX 78712. Current ad- dress: Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712 (arbogast@ices.utexas.edu). This author’s work was supported by U.S. National Science Foundation grants DMS-0713815 and DMS-0835745 and by a 2012 Moncrief Grand Challenge Faculty Award from the University of Texas at Austin. ‡ Corresponding author. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan, R.O.C. (huangcs@math.nsysu.edu.tw). This author’s work was supported in part under Taiwan National Science Council grant 99-2115-M-110-006-MY3. § Department of Mathematic and Physical Sciences, R.O.C. Air Force Academy, No. Sisou 1, Jieshou W. Rd., Gangshan Dist., Kaohsiung City 82047, Taiwan (hungch@math.nsysu.edu.tw). B447 Downloaded 09/09/12 to 140.117.35.119. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php