The use of Discontinuous Galerkin (DG) numerical 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 the 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 the moving-water steady flows. In this work, in the unified context consisting of third-order accurate DG-SWE schemes, a comparison between five numerical treatments of the bottom discontinuities is presented. We consider three widespread approaches that perform well if the motionless state has to be preserved. First, a simple technique, which consists in a proper initialization of the bed elevation that imposes 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 also consider two further approaches (both based on mechanical principles) 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 non-linear path. Several test cases are used to verify the accuracy, the well-balancing, the behavior in simulating a quiescent flow and the resolution in simulating unsteady flows 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.
A comparison between different approaches for the numerical treatment of bottom discontinuities in a DG perspective
CALEFFI, Valerio;VALIANI, Alessandro;
2015
Abstract
The use of Discontinuous Galerkin (DG) numerical 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 the 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 the moving-water steady flows. In this work, in the unified context consisting of third-order accurate DG-SWE schemes, a comparison between five numerical treatments of the bottom discontinuities is presented. We consider three widespread approaches that perform well if the motionless state has to be preserved. First, a simple technique, which consists in a proper initialization of the bed elevation that imposes 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 also consider two further approaches (both based on mechanical principles) 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 non-linear path. Several test cases are used to verify the accuracy, the well-balancing, the behavior in simulating a quiescent flow and the resolution in simulating unsteady flows 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.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
