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Journal article
Numerical algorithms for water waves with background flow over obstacles and topography
Advances in computational mathematics, v 48(4)
2022
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Abstract
We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).
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Details
- Title
- Numerical algorithms for water waves with background flow over obstacles and topography
- Creators
- David M. Ambrose - Drexel UniversityRoberto Camassa - University of North Carolina at Chapel HillJeremy L. Marzuola - University of North Carolina at Chapel HillRichard M. McLaughlin - University of North Carolina at Chapel HillQuentin Robinson - North Carolina Central UniversityJon Wilkening - University of California, Berkeley
- Publication Details
- Advances in computational mathematics, v 48(4)
- Publisher
- Springer US
- Grant note
- DMS-1716560 / national science foundation (https://doi.org/10.13039/100000001) DE-AC02-05CH11231 / u.s. department of energy (https://doi.org/10.13039/100000015) ONR N00014-18-1-2490 / office of naval research (https://doi.org/10.13039/100000006) DMS-1910824 / national science foundation (https://doi.org/10.13039/100000001) DMS-1909035 / national science foundation (https://doi.org/10.13039/100000001) DMS-1352353 / national science foundation (https://doi.org/10.13039/100000001) DMS-1907684 / national science foundation (https://doi.org/10.13039/100000001)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000822492300001
- Scopus ID
- 2-s2.0-85133703844
- Other Identifier
- 991019168586904721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied