Novel Schemes for Hyperbolic PDEs Using Osmosis Filters from Visual Computing
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- Hagenburg K., Breuß M., Weickert J., Vogel O. (2012) Novel Schemes for Hyperbolic PDEs Using Osmosis Filters from Visual Computing. In: Bruckstein A.M., ter Haar Romeny B.M., Bronstein A.M., Bronstein M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg
Recently a new class of generalised diffusion filters called osmosis filters has been proposed. Osmosis models are useful for a variety of tasks in visual computing. In this paper, we show that these filters are also beneficial outside image processing and computer graphics: We exploit their use for the construction of better numerical schemes for hyperbolic partial differential equations that model physical transport phenomena.
Our novel osmosis-based algorithm is constructed as a two-step, predictor-corrector method. The predictor scheme is given by a Markov chain model of osmosis that captures the hyperbolic transport in its advection term. By design, it also incorporates a discrete diffusion process. The corresponding terms can easily be identified within the osmosis model. In the corrector step, we subtract a stabilised version of this discrete diffusion. We show that the resulting osmosis-based method gives correct, highly accurate resolutions of shock wave fronts in both linear and nonlinear test cases. Our work is an example for the usefulness of visual computing ideas in numerical analysis.
Keywordsdiffusion filtering osmosis diffusion-advection drift-diffusion hyperbolic conservation laws finite difference methods predictor-corrector schemes stabilised inverse diffusion
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