Liu 200812

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High resolution computation of convection and diffusion is important in many applied partial differential equations, ranging from Euler, Navier-Stokes to Fokker-Planck equations. In this talk I shall present two novel numerical methods for problems involving these terms: i) the alternating evolution (AE) method for convection -- based on sampling of a refined description of the underlying equation on alternative grids, and ii) the direct discontinuous Galerkin (DDG) method for diffusion -- based on a novel numerical flux formula for the solution gradient. For each method I shall highlight the essential step --- a step in which the `physics' is incorporated into the method via the model refinement. The discretization of the refined model is then purely of numerical nature. Some numerical results will be presented to show the capacity of these methods.

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