Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

Abstract

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.

Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

“…Such a method of "extended" Lagrangian was efficiently used for dispersive models appearing both in classical and quantum fluids (e.g., Refs. [18][19][20][21][22][23][24]. map…”

Introduction to the Special Issue on Numerical Methods and Applications for Waves in Coastal Environments

“…In the work entitled Perfectly matched layers methods for mixed hyperbolic-dispersive equations [2], the authors deal with absorbing boundary conditions for the simulation of propagation of waves on bounded numerical domains. They study the stability of classical perfectly matched layer (PML) equations for different types of linear dispersive water wave models.…”

“…A review on the history and all fundamental aspects of the Boussinesq type (BT) models can be found in [3,15]. The works in the first category of this issue aims to answer some open problems concerning the discetisation of this type of models such as the absorbing boundary conditions for the simulation of waves on bounded domains [2] , the reduction of the computational cost which arise from the multi-step time schemes used to discretise the BT models [4] and the efficient discretisation of fully dispersive models with source terms [7].…”

The Conduit Equation: Hyperbolic Approximation and Generalized Riemann Problem

“…In [8,6,4,10,2,32], a general method of hyperbolic regularization of dispersive equations that are the Euler-Lagrange equations for a "master" Lagrangian has been proposed: the original high order derivative dispersive equations were approximated by a first order hyperbolic system of the Euler-Lagrange equations for a one or two parameter family of "extended" Lagrangians. The "master" Lagragian is obtained from the "extended" Lagrangian in some limit.…”

“…Unfortunately, it is not a new conservation law, but just a linear combination of (1) and (2). The equations (13) also admit a "symmetric" form.…”