Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries

Abstract

In this paper we consider the solution of the enhanced Boussinesq equations of Madsen and Sørensen (1992) [55] by means of residual based discretizations. In particular, we investigate the applicability of upwind and stabilized variants of continuous Galerkin finite element and Residual Distribution schemes for the simulation of wave propagation and transformation over complex bathymetries. These techniques have been successfully applied to the solution of the nonlinear Shallow Water equations (see e.g. Hauke (1998) [39] and Ricchiuto and Bollermann (2009) [61]). In a first step toward the construction of a hybrid model coupling the enhanced Boussinesq equations with the Shallow Water equations in breaking regions, this paper shows that equal order and even low order (second) upwind/stabilized techniques can be used to model non-hydrostatic wave propagation over complex bathymetries. This result is supported by theoretical (truncation and dispersion) error analyses, and by thorough numerical validation.

Introduction

The accurate simulation of nonlinear and non-hydrostatic wave propagation and transformation on complex bathymetries in the near shore region, up to the shoreline, plays a major role in coastal engineering. Numerical models for the applications involved benefit on one hand from the development of mathematical models with improved dispersion and shoaling characteristics, and, on the other, from the availability of accurate and stable discretizations of these equations.

Significant effort has been put in the last 20 years in development of systems of depth averaged equations which correctly reproduce the dispersion characteristics of wave propagation in the near shore region. Starting from the Boussinesq equations of Peregrine [56], several improved and enhanced Boussinesq models have been proposed over the years, including, among those having the largest impact in literature and the most recent ones, the enhanced equations of Madsen and Sørensen [55], the extended formulation of Nwogu [53], genuinely nonlinear Serre–Green–Naghdi equations [49], and nonlinear and non-hydrostatic higher order Shallow-Water type models [37]. These models have been obtained by retaining asymptotic behavior of the order of $\mathcal{O}({\mu }^2)$, μ being the ratio of water depth to wavelength. If $h_0$ is the value of a reference average depth, they give a correct description of the physics for values of the wave parameter $k{h}_0\approx 3\text{–}5$. More accurate models, including effects up to the $\mathcal{O}({\mu }^4)$ order have been proposed e.g. in [35].

Concerning the numerics used to solve these equations, the literature is full of promising schemes involving finite differences, finite volumes, or finite elements approaches. The major challenges that need to be dealt with are the approximation of the complex higher order derivative terms present in all non-hydrostatic depth-averaged models, and the accuracy requirements on the schemes in terms of low dispersion error. In addition, Boussinesq models can be coupled with the nonlinear Shallow-Water (NLSW) equations to model wave breaking [15], [68], [69], [67], [47]. While the mathematical character of the Boussinesq equations is (roughly) parabolic, the NLSW system is hyperbolic. As such it requires some degree of stabilization, e.g. in the form of some type of upwinding. A model coupling the Boussinesq system and the NLSW equations requires the underlying numerics used to robustly handle both the parabolic and purely hyperbolic limits.

The presence of higher order (third) partial derivatives has made the use of finite difference approximations appealing and quite popular (see e.g. [12], [33], [53], [34] to cite a few). The main drawback of the finite difference approach is the need of structured spatial meshes, even for irregular domains, and poor local mesh adaptivity potential (even tough hierarchical block structured multi-level approaches do exist, see e.g. [13]).

Fully unstructured solvers allowing for adaptive mesh refinement have been proposed, based either on the finite volume, or on the finite element approach. To the author's knowledge, genuinely multidimensional unstructured finite volume discretizations of enhanced Boussinesq equations have been actually proposed only in [46], [10], other works proposing some form of hybridization of finite volume/finite difference schemes on structured meshes or even in one space dimension (see e.g. [25], [16], [68] and references therein). The results presented in [46] are particularly encouraging, and the extension of the authors' model to wave breaking applications, presented at the Modeling the Earth system conference in Boulder [45], has shown the high potential of their approach. One criticism that can be made to the finite volume framework is that going beyond third order of accuracy might be quite hard, due to the necessity of introducing higher order multidimensional reconstructions for both the unknowns, and for the velocity divergence, to allow the discretization of the dispersive terms [46]. The advantage of the finite volume framework is of course an easy application of upwinding principles to properly handle the hyperbolic limit of the NLSW equations, and the use of well-established limiting techniques to avoid oscillations near bores and hydraulic jumps.

On the other hand, the finite element approximation gives a framework to naturally introduce higher order polynomial representation of the unknowns and of their derivatives, simply by handling these as auxiliary variables. The work of [30], [27] on discontinuous Galerkin approximations of enhanced Boussinesq models shows the potential in terms of accuracy of the finite element approach. Continuous Galerkin discretizations of Boussinesq models have been proposed by several authors. For example, in [21] a Taylor–Galerkin formulation for the Peregrine equations is discussed. More recently, a model based on Taylor–Galerkin time integration and the enhanced Madsen and Sørensen equations has been proposed in [54], using mixed approximation space. Standard Galerkin approximations are also discussed in [50], [72] (see also the PhD [71]). These contributions show results at least as good as those obtained by means of finite difference schemes, with the additional flexibility of a natural unstructured mesh formulation.

In this paper we want to add to this panorama an additional element, by analyzing and testing continuous finite element and residual based schemes which include some form of upwind stabilization, and which have already been shown to accurately and robustly handle the Shallow Water equations. In particular, we consider the Residual Distribution (RD) schemes, developed e.g. in [61], [60], [22], [17], and the upwind stabilized Galerkin scheme known as Streamline Upwind Petrov–Galerkin scheme (SUPG) of [39], [40], [43]. These schemes have shown very high potential in handling the NLSW, both in terms of preservation of physically relevant steady equilibria (well-balancedness), and in terms of a stable approximation of moving shorelines [61], [58], [39], [17]. For purely hyperbolic problems, it is known that, compared to finite differences, finite element schemes, and generally for residual based discretizations, have improved dispersion characteristics, due to the presence of a mass matrix.

While in the hyperbolic case this might seem like a drawback, in presence of mixed space and time derivatives, as in Boussinesq models, this gives an advantage, allowing to build discretizations that, on a reduced stencil, and even for low order interpolation (piecewise linear), yield dispersion properties similar to those of higher order finite difference schemes. Our aim is to analyze both theoretically and numerically second order upwind RD and SUPG discretizations for the enhanced Boussinesq equations of [55], and to asses their applicability to wave propagation. In the one-dimensional case, and for the linearized system, the paper presents a time-continuous error analysis based on a standard truncation error study of the finite difference form of the schemes, and a dispersion error analysis. The schemes are then thoroughly tested and compared to one another on one-dimensional benchmarks taken from the literature. Both the analytical and numerical results lead to the conclusion that the Petrov–Galerkin approach might be the best suited for this application. We thus consider a two-dimensional extension based on Petrov–Galerkin forms which, when considering the NLSW equations, give back the standard SUPG scheme, and the successful Multidimensional Upwind Residual Distribution scheme known as LDA scheme [22]. The results on well known two-dimensional benchmarks show that: on one hand the use of these schemes for wave propagation, on meshes with typical size comparable to that used by finite difference practitioners, is indeed feasible; on the other hand, that, compared to the standard SUPG stabilization, Multidimensional Upwinding leads to slightly less pronounced shoaling, and a lower content in higher harmonics. This work has to be understood as a first step toward the construction of a model including coupling with the NLSW equations to handle wave breaking and moving shorelines.

The structure of the paper is the following. In Section 2 we recall the basic form of the enhanced Boussinesq model of [55], and in Section 3 we present the schemes analyzed in one space dimension, including their implementation. Section 4 is devoted to a time continuous error analysis and comparison with second and higher order finite difference discretizations of the linearized equations. Section 5 discusses the issue of the initial and boundary conditions. The one-dimensional benchmarking is presented in Sections 6 and 7, which discuss in detail the CPU cost of the schemes considered and their comparison on several well established numerical tests. In Section 8, we discuss the extension of the schemes to two space dimension using a Petrov–Galerkin approach, and introducing two different generalizations of the upwind stabilization studied in 1d. Section 9 is devoted to the benchmarking of the Petrov–Galerkin schemes in two space dimensions. The paper is ended by a summary of the results and an outlook on ongoing work.