Discrete Transparent Boundary Conditions for the Linearized Green--Naghdi System of Equations

Abstract

In this paper, we introduce artificial boundary conditions for the linearized Green-Naghdi system of equations. The derivation of such continuous (respectively discrete) boundary conditions include the inversion of Laplace transform (respectively Z-transform) and these boundary conditions are in turn non local in time. In the case of continuous boundary conditions, the inversion is done explicitly. We consider two spatial discretisations of the initial system either on a staggered grid or on a collocated grids, both of interest from the practical point of view. We use a Crank Nicolson time discretization. The proposed numerical scheme with the staggered grid permits explicit Z-transform inversion whereas the collocated grid discretization do not. A stable numerical procedure is proposed for this latter inversion. We test numerically the accuracy of the described method with standard Gaussian initial data and wave packet initial data which are more convenient to explore the dispersive properties of the initial set of equations. We used our transparent boundary conditions to solve numerically the problem of injecting propagating (planar) waves in a computational domain.

Modeling shallow water waves

“…For transparent boundary conditions (which allow waves to cross the boundary of the computational domain without reflexion, see §3.3.3), the situation looks even more complicated. There are some results for the linear problem: for scalar equations (linear KdV or BBM for instance) [15,16] and for the linearization of (34) around the rest state [106]. The nonlinear case remains open.…”

“…generating and transparent) are much more complex and remain open. The case of transparent boundary conditions for the linearized SGN equations around the rest state (which are actually the same as the linearized Boussinesq equations around the rest state) has been addressed in [106].…”

A Domain Decomposition Method for Linearized Boussinesq-Type Equations

“…Therefore, we will need to derive four conditions. In this section, we follow the steps proposed in [5,7,8] to derive discrete TBC that are adapted to the discretized problem. We end this section with some numerical tests to analyze the efficiency of the obtained conditions.…”

“…Using the value z = −0.53753×h 0 (see [3]) and h 0 = 1, we have thath andh are both negative. Settingū = 0,h = 0,h = −ε a small parameter, g = 1 and h 0 = 1 leads to the formulation of the linearized Green-Naghdi equations, for which discrete transparent boundary conditions have been derived by Kazakova and Noble [5]. In this paper, we focus on the case whereū = 0 for the sake of simplicity, but the conclusions of Section 3 remain the same forū = 0.…”

“…The paper is organized as follows. In Section 2, we apply the study from [5,7,8] to the linearized Boussinesq equations and obtain discrete transparent boundary conditions. Then, we provide some numerical tests to evaluate the accuracy of such conditions before implementing them in a domain decomposition method.…”