O.: Novel schemes for hyperbolic PDEs using osmosis filters from visual computing

Abstract. Recently a new class of generalised diffusion filters called osmosis fil-ters 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 construc-tion 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 incorpo-rates a discrete diffusion process. The corresponding terms can easily be identi-fied within the osmosis model. In the corrector step, we subtract a stabilised ver-sion 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 comput-ing ideas in numerical analysis. Key words: diffusion filtering, osmosis, diffusion-advection, drift-diffusion, hy-perbolic conservation laws, finite difference methods, predictor-corrector schemes, stabilised inverse diffusion