New absorbing layers conditions for short water waves
Introduction
Many applications require to solve numerically dispersive wave problems in a domain which is much smaller than the physical one. A current method consists in applying a local absorbing boundary condition (ABC) on the exterior boundary which is used to limit the computational domain. The ABC should minimize spurious reflections when waves impinge on the exterior boundary which has no physical meaning. Obviously, the more efficient the ABC is, the more accurate the numerical solution will be. A lot of works have been dealing with the construction of ABCs that after discretization lead to a stable and accurate scheme and we refer to [2], [3] for discussion on related issues and for reviews on the subject. Most of the ABCs have been designed for either time-harmonic waves or for non-dispersive time-dependent waves and the use of ABCs in case of dispersive waves is much more difficult. A important example where dispersive waves must be considered is that of meteorological models which take into account the Earth’s rotation [4]. Recently, Joolen et al. [5] have developed a new numerical scheme including high-order ABCs for dispersive waves which are based on Higdon’s ABCs [6]. While it is possible to construct local ABCs easy to implement, their efficiency to damp spurious reflections often suffer from the corner problem. This difficulty can be overcome by using Perfectly Matched Layers just as was suggested by Bérenger [7], [8]. The very attractive property of the PML is it absorbs all the waves without spurious reflection and the corner problem is easily solved by a suitable fit of the layer parameters. Moreover, the coupling of the physical system with the PML condition is easy to handle numerically. Many works have been devoted to the design of PML for various applications and as far as the shallow-water equations are concerned, Navon et al. [9] have recently developed a split Perfectly Matched Layer (PML) scheme for the linearized system which is based on an explicit finite-difference discretization scheme. This PML requires a stabilization process involving a 9-point Laplacian filter to avoid the split PML supports unstable solutions. This question was formerly addressed for the linearized Euler equations in [10] and next in [11] for more general flows. Both in [10] and [11], the PML is obtained via a change of coordinate in the complex plane applied to the direction normal to the boundary. This amounts to replacing all the normal-derivatives in the Euler system by an operator which is still differential in the normal-direction but pseudo-differential in the other variables. In [1], Nataf has proposed another strategy which leads to the design of a stable PML for the Euler equation. The idea consists in applying the Smith factorization to the Euler equations in order to uncouple the propagative part of the solution from the transport one. Then the PML is constructed in such a way that only the modes that could produce reflections are damped. Thus the vorticity modes, which satisfy transparent conditions [12] on the computational boundary domain, are not damped and the resulting scheme seems to be stable as numerical experiments show.
In this paper, we propose a new PML formulation of the linearized shallow-water equations whose construction is based on the splitting of the primal system. Under the assumption the Coriolis force is constant, the depth h can be uncoupled from the velocity components by applying some elementary combinations which preserve the differential structure of the initial equations and the resulting formulation involves now a Klein–Gordon equation. The decomposition results in uncoupling the advective part of the wave from the vorticity (Section 2). After having remarked that the vorticity waves can be absorbed via an appropriate transparent condition and therefore do not need to be damped in the artificial layer, the PML condition is written by applying a complex coordinate change to the advective unknown only (Section 3). In Section 4 we discuss the practical handling of the method and we present the numerical scheme used to discretize the PML equations. Numerical results are presented in section 5. They confirm the new layer is perfectly matched to the physical domain and by considering long times of simulation, they seem to illustrate the long-time stability of the PML system.
Section snippets
Setting of the primal system
The shallow-water model contains some of the important dynamical features of the atmosphere and ocean and experience has shown that it is capable of describing main aspects of their motions. Let us consider a fluid with constant and uniform density. The height of the fluid surface above the reference level is . Even if h varies in space and time, we suppose that we can choose a characteristic value for the depth which is denoted by H. In the same way, we assume there exists a
The new PML model
Just as was previously mentioned, we focus on the description of a PML acting in the x-direction but what follows is not restrictive since it is sufficient to exchange x and U for y and V to obtain the corresponding PML in the y-direction.
The construction of the PML model is based on substituting only the x-derivatives acting on into the first equation of (7) by a PML-derivative. By this way, we aim at enforcing the stability of the PML model which is not guaranteed by substituting the x
Stability of the new PML
In this section, we assess the attenuation capability of the absorbing layer and for that purpose, we apply a modal analysis. A straightforward study consists in searching for under the formsolution to the problemfor any . Into the absorbing layer, the attenuation factor is related to the whose sign must be negative to strongly ensure absorption. The exponential
The PML system written as a first-order system
Systems (7), (10) both involve a second-order operator (L and ), whose numerical solution leads to the inversion of a full matrix. To avoid this difficulty, we have chosen, following Nataf [1], to rewrite these systems as systems requiring only the inversion of first-order operators. System (7) can obviously be rewritten as (2) and, by using in the first equation of (10), we obtain:Next, multiplying respectively by and the second and the third
Conclusion
We have constructed a new absorbing layer which is perfectly matched to the linearized shallow-water system. The construction process is applied to a second-order formulation of the equations which allows one to uncouple the height of water from the velocity field. By this way, only waves that can produce reflections at the interface between the physical domain and the absorbing one are damped. Then we have rewritten the PML equations as a first-order system by introducing auxiliary unknowns
Acknowledgments
These results were partially obtained during M. Tlemcani was visiting the team-project Magique-3d. The authors acknowledge the support by Egide Program. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of INRIA.
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2021, Journal of Computational PhysicsCitation Excerpt :Currently there is no reason to believe in the long-time stability of either 0-PML or α−PML in terms of analytical solutions. Nevertheless, a plane wave analysis with an infinite layer based on the slowness curves can be carried out without difficulty in order to show their stability at least for exponential modes as in [5–7]. – Comparison of FDTD and convolution analytical solutions
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2017, Journal of Computational PhysicsCitation Excerpt :Our idea is based on exploiting a very particular structure of the dispersion relation of the Maxwell equations with the plasma term, which allows us to split the original model into two different systems, to which we can apply two different kinds of the PMLs. This idea is new in the context of plasmas, however it bears some similarities to the method of [18], where a stable PML had been constructed for the shallow-water model. Indeed, this is not the first attempt of construction of stable perfectly matched layers in plasmas.
Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation
2015, Journal of Computational PhysicsCitation Excerpt :The first one is to map an unbounded domain to a bounded one, known as the Perfectly Matched Layer technique first introduced by Berenger [11] and later on used for many different partial differential equations. We specifically refer to e.g., [31,1,34,14,2,4,16,5,41,9] in the context of acoustic wave equations. The second approach, followed in this work, is to impose fictitious boundaries to truncate the domain of interest.
Absorbing layers for the Dirac equation
2015, Journal of Computational PhysicsCitation Excerpt :There are several ways to overcome such a difficulty. Modifications of the dispersion relation by a change of variables have been introduced in [7,20,24] for several equations such as the linearized Euler equation, resulting in a monotonic dispersion curve. Such an approach would require here a change of variables adapted to the discretization scheme.
A modified and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics
2013, Journal of Computational PhysicsCitation Excerpt :This so-called perfectly matched layer (PML) technique was first introduced by Berenger [11] using a splitting of the physical variables and considering a system of first order partial differential equations (PDEs) for electromagnetics. Since then, there has been much research work on this technique which subsequently was applied to different PDEs [2,5,15,31,37,3,38,41,43]. In the framework of time-harmonic wave propagation, the PML can be interpreted as a complex-valued coordinate stretching [42].
Perfectly matched layers for the heat and advection-diffusion equations
2010, Journal of Computational PhysicsCitation Excerpt :There is no need to approximate the square root of an operator by a partial differential operator. Since then, many works have been devoted to a better understanding of their principle and behavior see [10–20] to extensions to other geometries, see [21,22], or equations see [23–27]. In these works, the equations are hyperbolic and the need for a PML comes from the propagative modes that exist in the solution.