Abstract
We present the Galerkin boundary element method (BEM) for the numerical simulation of free-surface water waves in a model basin. In this work, as a first step we consider the linearized model of this time-dependent three-dimensional problem. After time discretization by an explicit Runge-Kutta scheme, the problem to be solved at each time step corresponds to the evaluation of a Dirichlet-to-Neumann map on the free surface of the domain. We use the Galerkin BEM for the approximate evaluation of the Dirichlet-to-Neumann map. To solve the resulting large, dense linear system, we use a data-sparse matrix approximation method based on hierarchical matrix representations. The proposed algorithm is quasi-optimal. Finally, some numerical results are given.
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References
Bebendorf, M.: ahmed. Another software library on hierarchical matrices for elliptic differential equations, Universität Leipzig, Fakultät für Mathematik und Informatik
Bebendorf, M.: Hierarchical Matrices. Springer, Heidelberg (2008)
Börm, S., Grasedyck, L.: HLib. A program library for hierarchical and H 2-matrices, Max Planck Institute for Mathematics in the Sciences, Leipzig
Cai, X., Langtangen, H.P., Nielsen, B.F., Tveito, A.: A finite element method for fully nonlinear water waves. J. Comput. Physics 143(2), 544–568 (1998)
Rjasanow, S., Steinbach, O.: The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering. Springer-Verlag New York, Inc. (2007)
Robertson, I., Sherwin, S.: Free-surface flow simulation using hp/spectral elements. J. Comput. Phys. 155(1), 26–53 (1999)
Tomar, S.K., van der Vegt, J.J.W.: A Runge-Kutta discontinuous Galerkin method for linear free-surface gravity waves using high order velocity recovery. Comput. Methods Appl. Meth. Engrg. 196, 1984–1996 (2007)
Westhuis, J.-H.: The Numerical Simulation of Nonlinear Waves in a Hydrodynamic Model Test Basin. PhD thesis, Universiteit Twente (2001)
Whitham, J.: Linear and Nonlinear Waves. John Wiley, New York (1974)
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Hofreither, C., Langer, U., Tomar, S. (2010). Boundary Element Simulation of Linear Water Waves in a Model Basin. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2009. Lecture Notes in Computer Science, vol 5910. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12535-5_14
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DOI: https://doi.org/10.1007/978-3-642-12535-5_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12534-8
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