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Upwind schemes for scalar advection-dominated problems in the discrete exterior calculus

: Griebel, Michael; Rieger, Christian; Schier, Alexander


Bothe, Dieter:
Transport processes at fluidic interfaces
Cham: Springer International Publishing, 2017 (Advances in mathematical fluid mechanics)
ISBN: 978-3-319-56601-6 (Print)
ISBN: 978-3-319-56602-3 (Online)
ISBN: 3-319-56601-6
Book Article
Fraunhofer SCAI ()

We present the discrete exterior calculus (DEC) to solve discrete partial differential equations on discrete objects such as cell complexes. To cope with advection-dominated problems, we introduce a novel stabilization technique to the DEC. To this end, we use the fact that the DEC coincides in special situations with known discretization schemes such as finite volumes or finite differences. Thus, we can carry over well-established upwind stabilization methods introduced for these classical schemes to the DEC. This leads in particular to a stable discretization of the Lie-derivative. We present the numerical features of this new discretization technique and study its numerical properties for simple model problems and for advection-diffusion processes on simple surfaces.