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.
Highlights
Linear and nonlinear wave equations are generally studied under the assumption that the solution is spatially periodic or decays to zero at infinity (Lax, 1976)
We performed a convergence study to demonstrate the effectiveness of the small-scale decomposition at removing stiffness from the evolution equations when the surface tension is large
We presented the results of a large-scale computation of a spatially quasi-periodic overturning water wave for which the wave peaks exhibit a wide array of dynamic behavior
Summary
Linear and nonlinear wave equations are generally studied under the assumption that the solution is spatially periodic or decays to zero at infinity (Lax, 1976). Beginning with Berenger (1994), a great deal of effort has been devoted to developing perfectly matched layer (PML) techniques for imposing absorbing boundary conditions over a finite computational domain to simulate wave propagation problems on unbounded domains. In many situations, assuming the waves decay to zero at infinity is not a realistic model. A large body of water such as the ocean is often covered in surface waves in every direction over vast distances. We formulate the initial value problem of the surface water wave equations in a spatially quasi-periodic setting, design numerical algorithms to compute such waves, and study their properties
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