Model order reduction strategies for weakly dispersive waves

We focus on the numerical modelling of water waves by means of depth averaged models. We consider in particular PDE systems which consist in a nonlinear hyperbolic model plus a linear dispersive perturbation involving an elliptic operator. We propose two strategies to construct reduced order models for these problems, with the main focus being the control of the overhead related to the inversion of the elliptic operators, as well as the robustness with respect to variations of the flow parameters. In a first approach, only a linear reduction strategies is applied only to the elliptic component, while the computations of the nonlinear fluxes are still performed explicitly. This hybrid approach, referred to as pdROM, is compared to a hyper-reduction strategy based on the empirical interpolation method to reduce also the nonlinear fluxes. We evaluate the two approaches on a variety of benchmarks involving a generalized variant of the BBM-KdV model with a variable bottom, and a one-dimensional enhanced weakly dispersive shallow water system. The results show the potential of both approaches in terms of cost reduction, with a clear advantage for the pdROM in terms of robustness, and for the EIMROM in terms of cost reduction

Upwind Stabilized Finite Element Modelling of Non-hydrostatic Wave Breaking and Run-up

In the following report a new methodology is presented to model the propagation, wave breaking and run-up of waves in coastal zones. We represent the different coastal phenomena through the coupling of non-linear shallow water equations with the extended Boussinesq equations of Madsen and Sørensen. Each of the involved equations has a major role in describing a particular physical behaviour of the wave: the latter equations permit to model the propagation, while the non-linear shallow water ones lead waves to locally converge into discontinuities. We start from the third-order stabilized finite element scheme for the Boussinesq equations, developed in a previous scientific work (Ricchiuto and Filippini, J.Comput.Phys. 2014) and develop a non-linear variant, and detach the dispersive from the shallow water terms. A shock-capturing technique based on local non-linear mass lumping that permits in the shallow water regions to degrade locally the scheme to a first-order one across bores (shocks) and dry fronts is proposed. As for the detection of the breaking fronts, the shallow water areas, this involves physics based breaking criteria. We present different definitions of the breaking criterion, including a local implementation of the convective criterion of (Bjørkavåg and H. Kalisch, Phys.Letters A 2011), and the hybrid models of (Kazolea et. al, J.Comput.Phys. 2014), and (Tonelli and Petti, J.Hydr.Res. 2011). The behavior of different breaking criteria is investigated on several cases for which experimental data are available.On décrit une approche pour la simulation de la propagation et déferlement des vagues en proche cote basée sur la couplage entre les équations de Boussinesq améliorées de Madsen and Sorensen, pour la propagation, et les équations Shallow Water, pour le déferlement et le runup. La contruction de ce modele hybride passe d'abord la proposition une variante non-linéaire du schéma élément finis stabilisé de (Ricchiuto and Filippini, J.Comput.Phys. 2014) capable de résoudre les chocs de maniere monotone. Cela est obtenu par un operateur locale de condensation de la matrice de masse qui réduit le schéma de (Ricchiuto and Filippini, J.Comput.Phys. 2014) au schéma de Roe classique. Le couplage entre le modèle Boussinesq et Shallow Water est en suite étudié. On considere différents critères physiques de détection de fronts déferlants. En particulier, on présente une implémentation numérique locale du critère convectif de (Bjorkavag and H. Kalisch, Phys.Letters A, 2011), qui est comparée au critères proposés dans (Kazolea et. al, J.Comput.Phys., 2014) et (Tonelli and Petti, J.Hydr.Res. 2011). Le modèle obtenu est validé sur des nombreux benchmarks avec données expérimentales

Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction

We present a novel approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit similar to the one of e.g. [20]. As in the reference, our scheme does not require any a-priori knowledge of the steady equilibrium, moreover it does not involve the explicit solution of any local auxiliary problem to approximate such equilibrium. The scheme is also arbitrarily high order, and verifies a continuous in time cell entropy equality. The latter becomes an inequality as soon as additional dissipation is added to the method. The method is constructed starting from a global flux approach in which an additional flux term is constructed as the primitive of the source. We show that, in the context of nodal spectral finite elements, this can be translated into a simple modification of the integral of the source term. We prove that, when using Gauss-Lobatto nodal finite elements this modified integration is equivalent at steady state to a high order Gauss collocation method applied to an ODE for the flux. This method is superconvergent at the collocation points, thus providing a discrete well-balanced property very similar in spirit to the one proposed in [20], albeit not needing the explicit computation of a local approximation of the steady state. To control the entropy production, we introduce artificial viscosity corrections at the cell level and incorporate them into the scheme. We provide theoretical and numerical characterizations of the accuracy and equilibrium preservation of these corrections. Through extensive numerical benchmarking, we validate our theoretical predictions, with considerable improvements in accuracy for steady states, as well as enhanced robustness for more complex scenario

Contributions to the development of residual discretizations for hyperbolic conservation laws with application to shallow water flows

In this work we review 12 years of developments in the field of residual based discretizations and their application to the solution of the shallow water equations. Fundamental concepts related to the topic are recalled and he construction of second and higher order schemes for steady problems is presented. The generalization to time dependent problems by means of multi-step implicit time integration, space-time, and genuinely explicit techniques is thoroughly discussed. Finally, the issues of C-property, super consistency, and wetting/drying are analyzed in this framework showing the power of the residual based approach

Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation, Stabilization, and Time-Stepping

In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov–Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge–Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel et al. (J Sci Comput 89(2):31, 2021. https://doi.org/10.1007/s10915-021-01632-7), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination

On wave breaking for Boussinesq-type models

International audienceWe consider the issue of wave breaking closure for Boussinesq type models, and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical set-up, and in particular to mesh size. For relatively classical choices of weakly dispersive propagation models, we compare two closure strategies. The first is the hybrid method consisting in suppressing the dispersive terms in breaking regions, as initially suggested by Tonelli and Petti in 2009. The second is an eddy viscosity approach based on the solution of a a turbulent kinetic energy. The formulation follows early work by O. Nwogu in the 90’s, and some more recent developments by Zhang and co-workers (Ocean Mod. 2014), adapting it to be consistent with the wave breaking detection used here. We perform a study of the behavior of the two closures for different mesh sizes, with attention to the possibility of obtaining grid independent results. Based on a classical shallow water theory, we also suggest some monitors to quantify the different contributions to the dissipation mechanism, differentiating those associated to the scheme from those of the partial differential equation. These quantities are used to analyze the dynamics of dissipation in some classical benchmarks, and its dependence on the mesh size. Our main results show that numerical dissipation contributes very little to the the results obtained when using eddy viscosity method. This closure shows little sensitivity to the grid, and may lend itself to the development and use of non-dissipative/energy conserving numerical methods. The opposite is observed for the hybrid approach, for which numerical dissipation plays a key role, and unfortunately is sensitive to the size of the mesh. In particular, when working, the two approaches investigated provide results which are in the same ball range and which agree with what is usually reported in literature. With the hybrid method, however, the inception of instabilities is observed at mesh sizes which vary from case to case, and depend on the propagation model. These results are comforted by numerical computations on a large number of classical benchmarks

Hyperbolic balance laws: residual distribution, local and global fluxes

This review paper describes a class of scheme named "residual distribution schemes" or "fluctuation splitting schemes". They are a generalization of Roe's numerical flux [61] in fluctuation form. The so-called multidimensional fluctuation schemes have historically first been developed for steady homogeneous hyperbolic systems. Their application to unsteady problems and conservation laws has been really understood only relatively recently. This understanding has allowed to make of the residual distribution framework a powerful playground to develop numerical discretizations embedding some prescribed constraints. This paper describes in some detail these techniques, with several examples, ranging from the compressible Euler equations to the Shallow Water equations

Analytical travelling vortex solutions of hyperbolic equations for validating very high order schemes

Testing the order of accuracy of (very) high order methods for shallow water (and Euler) equations is a delicate operation and the test cases are the crucial starting point of this operation. We provide a short derivation of vortex-like analytical solutions in 2 dimensions for the shallow water equations (and, hence, Euler equations) that can be used to test the order of accuracy of numerical methods. These solutions have different smoothness in their derivatives (up to $\mathcal C^\infty$) and can be used accordingly to the order of accuracy of the scheme to test

Construction of conservative PkPm space-time residual discretizations for conservation laws I : theoretical aspects

This paper deals with the construction of conservative high order and positivity preserving schemes for nonlinear hyperbolic conservation laws. In particular, we consider space-time Petrov-Galerkin discretizations inspired by residual distribution ideas and based on a PkPm polynomial approximations in space-time. The approximation is continuous in space and discontinuous in time so that one single space-time slab at the time can be dealt with. We show constructions involving linear high order and nonlinear schemes. Principles borrowed from the residual distribution approach, such as multidimensional upwinding and positivity preservation, are used to construct the Petrov-Galerkin test functions. The numerical results on one dimensional linear and nonlinear conservation laws show that higher accuracy and positivity are obtained uniformly with respect to the physical CFL number

Lifetime prediction of self-healing ceramic-matrix composites using a multi-physics image-based model

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