The development of efficient and accurate techniques for the simulation of advection remains one of the major challenges in the modelling of physical transport processes. The one-dimensional advection equation is frequently used to develop and test new advection schemes, which are then included in higher dimensional codes, such as full Navier-Stokes solvers. Additionally the solution of the one-dimensional advection diffusion equation is often required for water quality models associated with ground water and river flows [1]. Thecniques developed for the one dimensional advection equation will therefore have immediate application in water quality modelling, as well as providing an initial test-bed for multi-dimensional schemes.
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