We present a fully conservative, high-resolution, finite volume algorithm for advection-diffusion equations in irregular geometries. The algorithm uses a Cartesian grid in which some cells are cut by the embedded boundary. A novel feature is the use of a ``capacity function'' to model the fact that some cells are only partially available to the fluid. The advection portion then uses the explicit wave-propagation methods implemented in CLAWPACK, and is stable for Courant numbers up to 1. Diffusion is modelled with an implicit finite-volume algorithm. Results are shown for several geometries. Convergence is verified and the 1-norm order of accuracy is found to be- tween 1.2 and 2 depending on the geometry and Peclet number.
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