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Stable, high-order finite difference methods for nonlinear wave-structure interaction in a moving reference frame Bingham, Harry


This talk will focus on high-order finite difference methods for solving potential flow approximations of nonlinear surface waves interacting with marine structures. Of special interest is the loading and wave-induced response of sailing ships, where it is convenient to work in a reference frame which is translating at constant speed. This introduces non-linear convective terms into the free-surface boundary conditions which are found to require nonlinear numerical schemes to achieve robust and stable solutions. The work builds on the basic numerical solution strategy described in [1, 2]. Inspired by work reviewed for example by Shu [4], we have developed a simplified version of the Weighted Essentially Non-Oscillatory (WENO) scheme which is only slightly more diffusive than the equivalent order centered finite difference scheme. The equations are then put into Hamilton-Jacobi form, and the WENO convective approximations are combined using the Roe-Fix numerical flux proposed by Shu & Osher [5]. In contrast to a simple upwinding strategy, this approach is found to give accurate and stable solutions for all combinations of wave celerity and ship forward speed. To introduce the ship geometry into the numerical solution, we have developed an Immersed Boundary Method based on Weighted Least-Squares difference approximations. This scheme will be described and some preliminary results will be presented. Challenges with respect to tracking the body-free surface intersection line, and treating wave-breaking in a rational way will be raised for discussion. <br/> <br/> References: <ol> <li>H. B. Bingham and H. Zhang. On the accuracy of finite difference solutions for nonlinear water waves. J. Engineering Math., 58:211-228, 2007. </li> <li> A. P. Engsig-Karup, H. B. Bingham, and O. Lindberg. An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys., 228:2100-2118, 2009. </li> <li>A. P. Engsig-Karup, H. B. Bingham, O. Lindberg, and B. T. Paulsen. OceanWave3D; an efficient coastal engineering tool for nonlinear waves., 2015. </li> <li>C.-W. Shu. High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Review, 51(1):82-126, 2009. </li> <li>C.-W. Shu and S. Osher. Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for hyperbolic conservation laws. J. Comput. Phys., 77(2):439-471, 1989.</li> </ol>

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