A comparison between different approaches for the numerical treatment of bottom discontinuities in a DG perspective

Valerio Caleffi, Alessandro Valiani, Gang Li

Tuesday 30 june 2015

12:45 - 12:48h at Europe 1 & 2 (level 0)

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

Parallel session: Poster pitches: 5E. Engineering - Computional

The use of Discontinuous Galerkin (DG) schemes for the Shallow Water Equations (SWE) integration is greatly increased in the last decade. The efforts of many researchers were initially devoted to conceive techniques for the exact preservation of motionless state over non-flat bottom. Recently, such efforts are mainly oriented to the proper treatment of the bottom discontinuities and to the exact preservation of moving water steady flow. In this work, in the unified context represented by a third-order accurate DG- SWE scheme, a comparison between five numerical treatments of the bottom discontinuities is presented. We consider three widespread approaches that perform well if the motionless state have to be preserved. First, a simple technique, that consists in a proper initialization of the bed elevation that impose the continuity of the bottom profile is taken into account [Kesserwani and Liang, INT J NUMER METH ENG, 86, 47-69, 2011]. Then, we consider the hydrostatic reconstruction method [Audusse et al., SIAM J SCI COMPUT, 25, 2050-2065, 2004] and a path-conservative scheme based on a linear integration path [Parés, SIAM J NUMER ANAL, 44, 300-321, 2006]. We than consider two further approaches (the former characterized by a limited diffusion and the latter original) which are promising for the preservation of a moving-water steady state. A model is obtained modifying the hydrostatic reconstruction as suggested in [Caleffi and Valiani, ASCE JEM, 135(7), 684-696, 2009]. This method is characterized by a correction of the numerical flux based on the local conservation of the total energy. The last model is obtained improving the path-conservative scheme using a curvilinear path. Several test cases are used to verify the accuracy, the behavior in simulating a quiescent flow and the resolution of the models. A specific test case is also introduced to highlight the difference between the five schemes when a steady moving flow interacts with a bottom step.