A Splitting Method for Deep Water with Bathymetry

TL;DR

This paper introduces a new splitting numerical method for deep water wave models with bathymetry, proving its well-posedness, accuracy, and demonstrating its effectiveness through simulations of physical phenomena.

Contribution

It develops a novel splitting scheme for deep water models with bathymetry that avoids low pass filters and provides proven error estimates.

Findings

The scheme accurately approximates the system with order one in time.

Numerical experiments confirm the theoretical error estimates.

The method effectively simulates wave evolution over complex bottom topographies.

Abstract

In this paper we derive and prove the wellposedness of a deep water model that generalizes the Saut-Xu system for nonflat bottoms. Then, we present a new numerical method based on a splitting approach for studying this system. The advantage of this method is that it does not require any low pass filter to avoid spurious oscillations. We prove a local error estimate and we show that our scheme represents a good approximation of order one in time. Then, we perform some numerical experiments which confirm our theoretical result and we study three physical phenomena : the evolution of water waves over a rough bottom; the evolution of a KdV soliton when the shallowness parameter increases; the homogenization effect of rapidly varying topographies on water waves.