A Space-Time Hybridizable Discontinuous Galerkin Method for Linear Free-Surface Waves
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A Space-Time Hybridizable Discontinuous Galerkin Method for Linear Free-Surface Waves Giselle Sosa Jones1
· Jeonghun J. Lee2 · Sander Rhebergen1
Received: 17 June 2020 / Revised: 28 August 2020 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness and to obtain a priori error estimates, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. The analysis results are supported by numerical examples. Keywords Space-time · Hybridizable · Discontinuous Galerkin · Linear free-surface problem
1 Introduction In this paper, we are interested in the numerical solution of free-surface wave problems. In general, such problems are described by a set of partial differential equations that describe the movement of the fluid, and a set of boundary conditions that describe and determine the free-surface. These problems are particularly difficult to solve because the free-surface that defines the shape of the domain is part of the solution to the problem. To simplify the problem, we will assume an incompressible, inviscid, and irrotational fluid. Furthermore, we
SR gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada through the Discovery Grant program (RGPIN-05606-2015) and the Discovery Accelerator Supplement (RGPAS-478018-2015).
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Giselle Sosa Jones [email protected] Jeonghun J. Lee [email protected] Sander Rhebergen [email protected]
1
Department of Applied Mathematics, University of Waterloo, Waterloo N2L 3G1, Canada
2
Department of Mathematics, Baylor University, Texas 76798, USA 0123456789().: V,-vol
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Journal of Scientific Computing
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will assume that the wave displacement is small so that the free-surface boundary conditions may be linearized. In order to effectively model water waves, we require a stable and higher-order accurate numerical method to minimize numerical diffusion and dispersion. We remark that different numerical methods have been introduced to discretize free-surface wave problems, for example, boundary element methods have applied in [7,9,20]. See also the review papers [34,41]. In this work, however, we consider the discontinuous Galerkin (DG) method. The DG method for the spatial discretization combined with a higher-order accurate time stepping scheme has successfully been applied to linear water waves in [40,43]. In [43] they combined a second order accurate time stepping scheme with a higher-order accurate DG method for the spatial discretization. They proved stability and provided an a priori error analysis of their method. The primal
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