2020/01/31 by Jon Wilkening, Xinyu Zhao · 25 citations
Computer Science · Earth and Planetary Sciences · Engineering · Mathematics · Physics and Astronomy · #Coastal and Marine Dynamics #Discretization #Mathematical analysis #Mathematics #Ocean Waves and Remote Sensing #Wave and Wind Energy Systems #cs.NA #math.NA #msc:65E05 #msc:70K43 #msc:76B15 #msc:76M22 #physics.flu-dyn
paper · pdf · doi:10.1007/s00332-021-09689-2
published in Journal of Nonlinear Science 31(3) (Springer Science+Business Media) · 32 pages, 4 figures; expanded introduction, added two appendices
arxiv created 2021/02/10 · openalex publication_date 2021/04/20 · arxiv updated 2021/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.