Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes C. Mirabito a,⇑ , C. Dawson a , E.J. Kubatko b , J.J. Westerink c , S. Bunya d a Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA b Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus, OH 43210, USA c Environmental Fluid Dynamics Laboratories, Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN 46556, USA d Science and Safety Policy Research Division, Mitsubishi Research Institute, Inc., 2-3-6 O ¯ temachi, Chiyoda-ku, Tokyo 100-8141, Japan article info Article history: Received 26 February 2010 Received in revised form 10 August 2010 Accepted 12 August 2010 Available online xxxx Keywords: Shallow water Sediment transport Bed morphology Finite elements Discontinuous Galerkin abstract The shallow water equations are used to model large-scale surface flow in the ocean, coastal rivers, estu- aries, salt marshes, bays, and channels. They can describe tidal flows as well as storm surges associated with extreme storm events, such as hurricanes. The resulting currents can transport bed load and sus- pended sediment and result in morphological changes to the seabed. Modeling these processes requires tightly coupling a bed morphology equation to the shallow water equations. Discontinuous Galerkin finite element methods are a natural choice for modeling this coupled system, given the need to solve these problems on unstructured computational meshes, as well as the desire to implement hp-adaptivity for capturing the dynamic features of the solution. However, because of the presence of non-conservative products in the momentum equations, the standard DG method cannot be applied in a straightforward manner. To rectify this situation, we summarize and follow an extended approach described by Rheber- gen et al., which uses theoretical results due to Dal Maso et al. appearing in earlier work. In this paper, we focus on aspects of the implementation of the morphological model for bed evolution within the Advanced Circulation (ADCIRC) modeling framework, as well as the verification of the RKDG method in both h (mesh spacing) and p (polynomial order). This morphological model is applied to a number of coastal engineering problems, and numerical results are presented, with attention paid to the effects of h- and p-refinement in these applications. In particular, it is observed that for sediment transport, piecewise constant (i.e., finite volume) approximations of the bed are very over-diffusive and lead to poor sediment solutions. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction The modeling of flow and transport in coastal waters requires a detailed knowledge of winds, waves, currents, sediment transport and ultimately, the resulting morphological changes in the seabed that occur as a result of these processes. The erosion and deposition of bed sediment can have a major detrimental impact on the coastal population, infrastructure, and environment. For example, the US Army Corps of Engineers (USACE) maintains more than 12,000 miles of waterways for transportation, which carry approximately one-sixth of the US inter-city freight [1]. Maintenance of these waterways through dredging and backfilling operations represents a significant cost to the USACE as well as other agencies and indus- tries. As another example, the structural integrity of bridges, levees, and piers can be compromised by excessive scour of the seabed around the structure. Also, dune, barrier island, and channel erosion during a hurricane leads to the removal of major flow controls, which significantly affects inland inundation. In addition to these infrastructure-related issues, there exists a host of environmental concerns, such as beach erosion and the transport of contaminants with sediment, which may actually act as a source or sink for con- taminants depending on the surrounding physico-chemical condi- tions [2]. Collectively, the various fluid flow and transport processes that lead to morphological changes in the seabed form an interdepen- dent physical system in which the fluid motion, due to both waves and currents, drives the transport of sediment, which dictates the morphological evolution of the seabed. In turn, the fluid motion it- self is then directly affected by the morphological changes in the bed that it induces. Comprehensive modeling of these processes in the coastal zone presents several challenges and open questions. Most existing hydrodynamic models use a fixed-bed approach; that is, the bottom boundary of the seabed is not allowed to evolve 0045-7825/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2010.08.004 ⇑ Corresponding author. Tel.: +1 512 471 1721. E-mail addresses: mirabito@ices.utexas.edu (C. Mirabito), clint@ices.utexas.edu (C. Dawson), kubatko.3@osu.edu (E.J. Kubatko), jjw@nd.edu (J.J. Westerink), sbunya@gmail.com (S. Bunya). Comput. Methods Appl. Mech. Engrg. xxx (2010) xxx–xxx Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma Please cite this article in press as: C. Mirabito et al., Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes, Comput. Methods Appl. Mech. Engrg. (2010), doi:10.1016/j.cma.2010.08.004