A Domain Decomposition Method for Linearized Boussinesq-Type Equations

sci, Joao Guilherme Caldas Steinstraesser

doi:10.4208/jms.v52n3.19.06

JMS

2019

DOI: 10.4208/jms.v52n3.19.06

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A Domain Decomposition Method for Linearized Boussinesq-Type Equations

Joao Guilherme Caldas Steinstraesser sci¹

Abstract: In this paper, we derive discrete transparent boundary conditions for a class of linearized Boussinesq equations. These conditions happen to be non-local in time and we test numerically their accuracy with a Crank-Nicolson time-discretization on a staggered grid. We use the derived transparent boundary conditions as interface conditions in a domain decomposition method, where they become local in time. We analyze numerically their efficiency thanks to comparisons made with other interface conditions.

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Cited by 2 publications

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Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

2022

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Absorbing boundary conditions are important when one simulates the propagation of waves on a bounded numerical domain without creating artificial reflections. In this paper, we consider various hyperbolic-dispersive equations modeling water wave propagation. A typical example is the Korteweg-de Vries equation (1)In the case of linearized equations, some progress was recently done for one dimensional scalar dispersive equations by using discrete transparent boundary conditions. However a generalization of this approach to multi-dimensional setting is not obvious. In this paper, we consider the alternative perfectly matched layer (PML) approach for the linearized Korteweg-de Vries equation:(2)where U ∈ R denotes a reference speed. We first propose a direct perfectly matched layer approach and study the stability of the modified system. These equations are not always stable, the main obstruction being the classical condition v g (k)v φ (k) ≥ 0 found in the literature on PML [3] that we recover in our analysis. Then, we introduce a hyperbolic system with a source term that is an approximation of the Korteweg-de Vries equations. In this case, the complete PML equations are not, again, completely stable. However, a version of the PML equations for this system derived without the source term is found to be stable and can absorb outgoing waves although it may creates reflections as it is not perfectly matched. Finally, we consider the BBM-Boussinesq system that models bi-directional waves at the surface of an inviscid fluid layer. The dispersive properties for a subclass of physically relevant models are better suited for PML techniques since the condition v g (k)v φ (k) ≥ 0 is always satisfied. We show that the PML equations are always stable in this case. We illustrate numerically the absorbing and stability properties of these PML models and provide also KdV type simulation by choosing properly initial data.

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Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

2022

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General boundary conditions for a Boussinesq model with varying bathymetry

Lannes,

Rigal

2024

Stud Appl Math

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This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

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A Domain Decomposition Method for Linearized Boussinesq-Type Equations

Cited by 2 publications

References 10 publications

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

General boundary conditions for a Boussinesq model with varying bathymetry

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