Adaptive, multiresolution, Discontinuous Galerkin Shallow Water modelling

Daniel Caviedes-Voullième, Georges Kesserwani, Nils Gerhard, Siegfried Müller

Thursday 2 july 2015

11:45 - 12:00h at Oceania (level 0)

Themes: (T) Water engineering, (ST) Computational methods

Parallel session: 11E. Engineering - Computational

Shallow water modelling poses challenges not only on the formulation of numerical schemes, but also in modelling and solving the different spacial and temporal scales involved. Many applications of shallow water modelling including flood simulation and coastal simulation often require to represent and solve very large domains, while being able to include small scale topography features and track moving and transient flow features, such as shocks and wet/dry fronts. Adaptive meshes can provide the optimal solution for the multiresolution problem that shallow water problems pose, and indeed a number of strategies have been implemented and tested. However, in most cases, adaptivity is governed by extrinsic, user-controlled criteria which is not necessarily optimal from neither the accuracy or efficiency point of views. This work presents an adaptive, multiresolution Shallow Water solver, based on the Discontinuous Galerkin method and multiwavelets (MWDG). The mathematical properties of the MWDG scheme allow for it to automatically adapt over a large number of resolution levels using information from the solution itself and a single user-specified parameter. This is done via the filter action that multiwavelets can exert on the DG modal decomposition of the partial differential equations. We present the fundamentals of the SW-MWDG formulation, show that desirable properties for Shallow Water modelling are enforced and guaranteed under adaptive meshes (well balancing and positivity preserving) and include benchmarking cases.