High-order modeling of long dispersive water waves

February 28, 2025

TIME: 3:30 PM

LOCATION: GMCS 314

SPEAKER: Christos Papoutsellis, University of California, San Diego

ABSTRACT:

Accurately predicting the intricate dynamics of water waves is essential for advancing our understanding of the marine environment. The modeling of long waves plays a critical role in applications such as tsunami prediction, extreme wave simulations, and wave interactions with bathymetry and coastal structures. As water wave models continue to evolve, they aim to accommodate larger spatial domains, longer simulation times, and greater physical accuracy. These models are formulated as Partial Differential Equations (PDE) and aim to replace the original free-boundary hydrodynamic problem by a more numerically tractable system while preserving key physical properties. Despite significant progress and the numerous models proposed, further refinements are necessary to more accurately capture strong nonlinear and dispersive effects while ensuring numerical stability.

In this talk, we will examine water waves governed by the free-surface Euler equations under the assumption of irrotational flow. I will introduce a modeling approach to derive reduced model equations from the variational principle of the original problem, employing series representations to approximate the velocity potential. Drawing inspiration from long-wave theory, we construct an ansatz consisting of prescribed vertical polynomials with unknown horizontal functional coefficients. The resulting family of model equations retains a canonical nonlocal Hamiltonian structure, closely mirroring that of the original problem. Specifically, wave evolution is described by a system of two nonlinear evolution PDEs coupled with a system of linear elliptic PDEs on the horizontal plane. This family of model equations provides high-order long-wave approximations for the water-wave problem while containing spatial derivatives of at most second order, distinguishing it from asymptotic methods involving higher-order mixed spatio-temporal derivatives. I will highlight connections between this new model and both classical and recent asymptotic models, and present numerical solutions that capture strongly nonlinear and dispersive wave dynamics.

VIDEO: