A three dimensional Dirichlet-to-Neumann operator for waves over topography

TL;DR

This paper develops a three-dimensional Dirichlet-to-Neumann operator for water waves over complex topographies, simplifying 3D wave modeling to 2D surface computations and validating it through numerical simulations.

Contribution

It introduces a novel 3D DtN operator for waves over non-smooth topographies, including a specialized Galerkin method for the topographic component.

Findings

The DtN operator accurately models waves over large amplitude, rapidly varying topographies.

Numerical simulations confirm the operator's effectiveness in complex topographic scenarios.

Benchmarking against conformal mapping demonstrates the method's precision.

Abstract

Surface water waves are considered propagating over highly variable non-smooth topographies. For this three dimensional problem a Dirichlet-to-Neumann (DtN) operator is constructed reducing the numerical modeling and evolution to the two dimensional free surface. The corresponding Fourier-type operator is defined through a matrix decomposition. The topographic component of the decomposition requires special care and a Galerkin method is provided accordingly. One dimensional numerical simulations, along the free surface, validate the DtN formulation in the presence of a large amplitude, rapidly varying topography. An alternative, conformal mapping based, method is used for benchmarking. A two dimensional simulation in the presence of a Luneburg lens (a particular submerged mound) illustrates the accurate performance of the three dimensional DtN operator.