A modified Galerkin/finite element method for the numerical solution of the Serre‐Green‐Naghdi system

Mitsotakis, Dimitrios; Synolakis, Costas E.; McGuinness, Mark

doi:10.1002/fld.4293

Cited by 31 publications

(54 citation statements)

References 61 publications

(157 reference statements)

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“…x " uL h tuu`"ǫh 3 uux ‰ x " mG h tmu`'boundary terms', so the kinetic energy part of the Hamiltonian density (2.23) can be replaced by 1 2 mG h tmu. for the temporal discretisation, as described and analysed in [22] for a flat bottom and in [23] for varying bottoms. This method can perform really well for smooth solutions due to its conservative properties.…”

Section: Numerical Resultsmentioning

confidence: 99%

Hamiltonian regularisation of shallow water equations with uneven bottom

Clamond¹,

Dutykh²,

Mitsotakis³

2019

J. Phys. A: Math. Theor.

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The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called Hamiltonian regularisation for nonlinear shallow water and isentropic Euler equations. The characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or dispersion. The regularised system possesses a Hamiltonian structure and, thus, formally preserves the corresponding energy functional. In the present article we generalise this approach to shallow water waves over general, possibly time-dependent, bottoms. The proposed system is solved numerically with continuous Galerkin method and its solutions are compared with the analogous solutions of the classical shallow water and dispersive Serre-Green-Naghdi equations. The numerical results confirm the absence of dispersive and dissipative effects in presence of bathymetry variations.

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Section: Numerical Resultsmentioning

confidence: 99%

Hamiltonian regularisation of shallow water equations with uneven bottom

Clamond¹,

Dutykh²,

Mitsotakis³

2019

J. Phys. A: Math. Theor.

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“…Section 5.2 of [77]. Recently, [71] demonstrated that the fully nonlinear Serre-Green-Nagdhi (SGN) equations simulate this experiment with higher accuracy than weakly nonlinear models. As will be demonstrated below, our method, being fully nonlinear, captures nicely this phenomenon, as expected.…”

Section: Reflection Of Shoaling Solitary Waves On a Vertical Wall At mentioning

confidence: 90%

“…The extension of the region of validity of these models and their numerical implementation has been the subject of a great deal of work. Restricting attention to models that take into account an uneven seabed, we mention, for instance, i) the Boussinesq-type models [1], [2], [3], [4], [5], [6], [7], [8], [9], ii) the two layer models [10], [11], and iii) the Green-Nagdhi equations [12], [13], [14]. More details and references can be found in Chapter 7 of [15] and in the review articles [16], [17].…”

Section: Introductionmentioning

confidence: 99%

Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to solitary wave problems over bathymetry

Papoutsellis

Charalampopoulos

Athanassoulis

2018

European Journal of Mechanics - B/Fluids

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1. Introduction 2. Formulation of the problem 2.1 Classical formulation of the problem 2.2 The Hamiltonian Coupled-Mode Theory 2.2.1.. Exact vertical series expansion of the wave potential 2.2.2 The Hamiltonian coupled-mode system 2 6. Interaction of solitary waves with bathymetry and vertical walls 6.1 Reflection of solitary waves on a vertical wall 6.2 Shoaling of solitary waves over a plane beach 6.3 Reflection of shoaling solitary waves on a vertical wall at the end of a sloping beach 6.4 Propagation of a solitary wave over a sinusoidal patch 6.5 Transformation of a solitary wave over a 3D bathymetry with banks and trenches 7. Discussion and conclusions Acknowledgments Appendix A: An outline of the derivation of the Hamiltonian Coupled Mode System Appendix B: Calculation of the basic vertical integrals (a) Vertical integrals of the form ( ; ) m J s Z and ( ; ) m J s W (b) Vertical integrals of the form ( ; ) n m J s Z Z for 0 ,1, 2 s = (c) Vertical integrals of the form ( ; ) n m J s W Z for 0 ,1 s = Appendix C. Fast and accurate calculation of local wavenumbers ( , ) n k t x Appendix D: Finite-difference linear system for the substrate problem Eqs. (17a,b) Appendix E. Description of the 3D bathymetry References List of abbreviationsBEM boundary element method CMS coupled-mode system DNM direct numerical methods DtN Dirichlet-to-Neumann (operator) FD finite difference FDM finite difference method FEM finite element method HCMS Hamiltonian coupled-mode system HCMT Hamiltonian Coupled-Mode Theory NLPF nonlinear potential flow NR Newton-Raphson (method) RK Runge-Kutta (method) SGN Serre-Green-Nagdhi (equations) AbstractThis paper deals with the implementation of a new, efficient, non-perturbative, Hamiltonian coupled-mode theory (HCMT) for the fully nonlinear, potential flow (NLPF) model of water waves over arbitrary bathymetry Papoutsellis and Athanassoulis (2017) (arxiv.org/abs/1704.03276). Applications considered herein concern the interaction of solitary waves with bottom topographies and vertical walls both in two-and three-dimensional environments. The essential novelty of HCMT is a new representation of the Dirichlet-to-Neumann operator, which is needed to close the Hamiltonian evolution equations. This new representation emerges from the treatment of the substrate kinematical problem by means of exact semi-separation of variables in the instantaneous, irregular, fluid domain, established recently by Athanassoulis & Papoutsellis (2017) (https://doi.org/10.1098/rspa.2017.0017). The HCMT ensures an efficient dimensional reduction of the exact NLFP, being able to treat an arbitrary bathymetry as simply as the flat-bottom case, without domain transformation. A key point for the efficient implementation of the method is the fast and accurate evaluation of the space-time varying coefficients appearing in some of its equations. In this paper, all varying coefficients are calculated analytically, resulting in a refined version of the theory, characterized by improved accuracy at significantly reduced computationa...

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“…It would be instructive to calculate the rate of convergence of the iteration (5.24). Table 1 lists values for the rates of convergence [8] …”

Section: Remarkmentioning

confidence: 99%

A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

Jang

2017

J Sci Comput

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This paper begins with a question of existence of a regular integral equation formalism, but different from the existing usual ones, for solving the standard Boussinesq's equations for variable water depth (or Peregrine's model). For the question, a pseudo-water depth parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118-138, 2017), is introduced to alter the standard Boussinesq's equations into an integral formalism. This enables us to construct a regular (nonlinear) integral equations of second kind (as required), being equivalent to the standard Boussinesq's equations (of Peregrine's model). The (constructed) integral equations are, of course, inherently different from the usual integral equation formalisms. For solving them, the successive approximation (or the fixed point iteration) is applied (Jang 2017), whereby a new iterative formula is immediately derived, in this paper, for numerical solutions of the standard Boussinesq's equations for variable water depth. The formula, semi-analytic and derivative-free, is shown to be useful to observe especially the nonlinear wave phenomena of shallow water waves on a beach. In fact, a numerical experiment is performed on a solitary wave approaching a sloping beach. It shows clearly the main feature of nonlinear wave characteristics, which has reached good agreement with the known (numerical) solutions. Hence, while being theoretical but fundamental in nonlinear computational partial differential equations, the question raised in the study may be solved.Keywords A regular integral equation formalism · The standard Boussinesq's equations for variable water depth · A pseudo-water depth parameter · The successive approximation

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A modified Galerkin/finite element method for the numerical solution of the Serre‐Green‐Naghdi system

Cited by 31 publications

References 61 publications

Hamiltonian regularisation of shallow water equations with uneven bottom

Hamiltonian regularisation of shallow water equations with uneven bottom

Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to solitary wave problems over bathymetry

A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

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