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 multiplevalued velocity potentials but singlevalued stream functions. We prove that the resulting secondkind Fredholm integral equations are invertible, possibly after a physically motivated finiterank correction. In an anglearclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravitycapillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually selfintersects 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).
more » « less NSFPAR ID:
 10369687
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Advances in Computational Mathematics
 Volume:
 48
 Issue:
 4
 ISSN:
 10197168
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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