Discrete transparent boundary conditions for the mixed KDV-BBM equation

TL;DR

This paper develops stable, general strategies for implementing transparent boundary conditions in simulations of the mixed KDV-BBM equation, improving accuracy for water wave modeling over large times.

Contribution

It introduces a novel, stable approach for computing nonlocal boundary operators and asymptotic methods for large-time simulations in water wave models.

Findings

Methods accurately simulate Gaussian and wave packet initial data.

Proposed strategies improve stability and generality of boundary conditions.

Effective for large time water wave simulations.

Abstract

In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomor-phic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.