Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlin-ear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data

Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green‐Naghdi equations

This paper presents a multiresolution discontinuous Galerkin scheme for the adaptive solution of Boussinesq‐type equations. The model combines multiwavelet‐based grid adaptation with a discontinuous Galerkin (DG) solver based on the system of fully nonlinear and weakly dispersive Green‐Naghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using multiwavelets on a hierarchy of nested grids to improve the efficiency of the reference DG scheme on a uniform grid by computing on a locally refined adapted grid. This way the local resolution level will be determined by manipulating multiwavelet coefficients controlled by a single user‐defined threshold value. The proposed adaptive multiwavelet discontinuous Galerkin solver for GN equations (MWDG‐GN) is assessed using several benchmark problems related to wave propagation and transformation in nearshore areas. The numerical results demonstrate that the proposed scheme retains the accuracy of the reference scheme, while significantly reducing the computational cost

High-Order Numerical Methods in Lake Modelling

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

Discontinuous Galerkin methods for resolving non linear and dispersive near shore waves

textNear shore hydrodynamics has been an important research area dealing with coastal processes. The nearshore coastal region is the region between the shoreline and a fictive offshore limit which usually is defined as the limit where the depth becomes so large that it no longer influences the waves. This spatially limited but highly energetic zone is where water waves shoal, break and transmit energy to the shoreline and are governed by highly dispersive and non-linear effects. An accurate understanding of this phenomena is extremely useful, especially in emergency situations during hurricanes and storms. While the shallow water assumption is valid in regions where the characteristic wavelength exceeds a typical depth by orders of magnitude, Boussinesq-type equations have been used to model near-shore wave motion. Unfortunately these equations are complex system of coupled non-linear and dispersive differential equations that have made the developement of numerical approximations extremely challenging. In this dissertation, a local discontinuous Galerkin method for Boussinesq-Green Naghdi Equations is presented and validated against experimental results. Currently Green-Naghdi equations have many variants. We develop a numerical method in one horizontal dimension for the Green-Naghdi equations based on rotational characteristics in the velocity field. Stability criterion is also established for the linearized Green-Naghdi equations and a careful proof of linear stability of the numerical method is carried out. Verification is done against a linearized standing wave problem in flat bathymetry and h,p (denoted by K in this thesis) error rates are plotted. The numerical method is validated with experimental data from dispersive and non-linear test cases.Aerospace Engineerin

Numerical modelling in a multiscale ocean

Systematic improvement in ocean modelling and prediction systems over the past several decades has resulted from several concurrent factors. The first of these has been a sustained increase in computational power, as summarized in Moore\u27s Law, without which much of this recent progress would not have been possible. Despite the limits imposed by existing computer hardware, however, significant accruals in system performance over the years have been achieved through novel innovations in system software, specifically the equations used to represent the temporal evolution of the oceanic state as well as the numerical solution procedures employed to solve them. Here, we review several recent approaches to system design that extend our capability to deal accurately with the multiple time and space scales characteristic of oceanic motion. The first two are methods designed to allow flexible and affordable enhancement in spatial resolution within targeted regions, relying on either a set of nested structured grids or, alternatively, a single unstructured grid. Finally, spatial discretization of the continuous equations necessarily omits finer, subgrid-scale processes whose effects on the resolved scales of motion cannot be neglected. We conclude with a discussion of the possibility of introducing subgrid-scale parameterizations to reflect the influences of unresolved processes

Stationary shock-like transition fronts in dispersive systems

International audienceWe show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine–Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not travelling waves and apparently require for their description more complex anzats, evolve into travelling wave type jump discontinuities

Numerical methods for Shallow Water Equations

Στο πρώτο κεφάλαιο διατυπώνονται οι εξισώσεις του Euler για κύματα επιφανείας ενός τέλειου ρευστού (π.χ. νερού) σε διδιάστατο κυματοδηγό πεπερασμένου βάθους με μεταβλητή τοπογραφία πυθμένα. Οι εξισώσεις γράφονται σε αδιάστατη, κανονικοποιημένη μορφή με παραμέτρους κλίμακας ε=α₀/λ₀, μ=(D₀/λ₀)², όπου α₀ τυπικό πλάτος των κυμάτων, λ₀ τυπικό μήκος κύματος, και D₀ το μέσο βάθος πυθμένα. Από τις εξισώσεις του Euler παράγονται προσεγγιστικά, απλούστερα μοντέλα για την περιγραφή της κίνησης μη γραμμικών, διασπειρομένων κυμάτων επιφανείας σε δύο κατευθύνσεις με μεγάλο μήκος κύματος σχετικά με το μέσο βάθος του πυθμένα, δηλ. για τα οποία μ≪1. Το βασικό μοντέλο είναι οι εξισώσεις Serre-Green-Naghdi (SGN) με μεταβλητό πυθμένα, από τις οποίες παράγονται εν συνεχεία τρία απλούστερα μοντέλα σε ειδικές περιοχές των παραμέτρων κλίμακας: Α) το κλασσικό σύστημα Boussinesq με μεταβλητό πυθμένα γενικής τοπογραφίας (CBs), στο οποίο ε=O(μ) και β=Ο(1), όπου β=B/D₀, με Β να είναι τυπικό μέγεθος της μεταβολής του πυθμένα. Β) το κλασσικό σύστημα Boussinesq με ασθενή μεταβολή του πυθμένα, β=O(ε), (CBw). Γ) το σύστημα των εξισώσεων ρηχών υδάτων (SW) για το οποίο μ=0 και γενικά ε=O(1). Το δεύτερο κεφάλαιο αφορά την αριθμητική ανάλυση προβλημάτων αρχικών και συνοριακών συνθηκών (α.σ.σ) για τα συστήματα (CBs), (CBw), σε πεπερασμένο διάστημα με u=0 στο σύνορο. Ύστερα από ανασκόπηση της θεωρίας ύπαρξης-μοναδικότητας των λύσεων των προβλημάτων αυτών, τα συστήματα διακριτοποιούνται ως προς την χωρική συνιστώσα με την συνήθη μέθοδο Galerkin-πεπερασμένων στοιχείων, και εκτιμάται το σφάλμα της ημιδιακριτοποίησης αυτής στον L²×H¹. Η εκτίμηση του σφάλματος επιβεβαιώνεται αριθμητικά. Εξετάζονται και τέτοιες αριθμητικές μέθοδοι για την περίπτωση απορροφητικών συνθηκών στο σύνορο. Τέλος εξετάζονται υπολογιστικά, με βάση κυρίως το μοντέλο (CBs), φαινόμενα που αφορούν μεταβολές που υφίσταται ένα αρχικά μοναχικό κύμα όταν κινείται σε περιβάλλον με πυθμένα μεταβλητής τοπογραφίας. Στο τρίτο κεφάλαιο εξετάζεται το σύστημα (SW) με μεταβλητό πυθμένα υπό την προϋπόθεση ότι έχει ομαλές λύσεις. Αποδεικνύονται εκτιμήσεις σφαλμάτων των μεθόδων Galerkin-ΠΣ στον χώρο L²×L² για το πρόβλημα α.σ.σ. με u=0 στα άκρα πεπερασμένου διαστήματος, καθώς και με χαρακτηριστικές συνοριακές συνθήκες απορρόφησης για υπερκρίσιμες ή υποκρίσιμες ροές. Εξετάζεται υπολογιστικά η ικανότητα της αριθμητικής μεθόδου να προσεγγίζει λύσεις σταθερής μορφής και λύσεις της μορφής «ηρεμουσών ροών» όταν το σύστημα γραφτεί σε μορφή νόμου ισορροπίας. Στο τέταρτο κεφάλαιο εξετάζεται η ασυνεχής μέθοδος Galerkin-ΠΣ (DG) για το σύστημα (SW), γραμμένο σε μορφή νόμου ισορροπίας. Γίνεται ανασκόπηση των μεθόδων RKDG για υπερβολικά συστήματα νόμων διατήρησης σε μία διάσταση και εξετάζονται τεχνικές περιορισμού κλίσης. Κατόπιν εξετάζονται οι μέθοδοι RKDG για (SW) με μεταβλητό πυθμένα. Εξετάζονται θέματα και αλγόριθμοι καλής εξισορρόπησης, διατήρησης του μη-αρνητικού βάθους της στήλης νερού όταν ο πυθμένας πλησιάζει την ελεύθερη επιφάνεια, περιορισμού της κλίσης σε περίπτωση ασυνεχειών κ.α. Τέλος παρατίθεται μια σειρά προβλημάτων δοκιμής τα οποία ο αλγόριθμος προσεγγίζει με μεγάλη ακρίβεια.In the first chapter we state the Euler equations describing surface waves of an ideal fluid (water) in a two-dimensional waveguide of finite depth with variable bottom topography. The equations are written in nondimensional, scaled form using the scaling parameters ε=α₀/λ₀, μ=(D₀/λ₀)², where α₀ is a typical wave amplitude, λ₀ a typical wavelength, and D₀ an average bottom depth. From the Euler equations we derive a series of simple, approximate, models, that describe two-way propagation of nonlinear, dispersive surface waves in one dimension, that are long compared to the average bottom depth, i.e. satisfy μ≪1. The basic model are the Serre-Green-Naghdi (SGN) equations, from which three simpler mathematical models follow in specific regimes of scaling parameters: Α) the Classical Boussinesq system with variable bottom of general topography (CBs), in which ε=O(μ) and β=Ο(1), where β=B/D₀, with Β a typical bottom topography variation. B) the Classical Boussinesq system with weakly varying bottom, i.e. β=O(ε), (CBw). C) The system of shallow water equations (SW), where μ=0 and in general ε=O(1). The second chapter concerns the numerical analysis of initial and boundary value problems (ibvp’s) for the (CBs) and (CBw) systems in a finite interval with u=0 at the boundary. After a review of their theory of existence-uniqueness of solutions, the systems are discretized in space by the standard Galerkin-finite element method, and the semidiscretization error is estimated in L²×H¹. This estimate is verified by numerical experiments. We also examine Galerkin-FE methods for (CBw), (CBs), and (SW) with absorbing boundary conditions. Finally we study numerically, using mainly (CBs), changes that an initial solitary wave undergoes when moving into a region of variable bottom topography. In the third chapter we consider the (SW) with variable bottom, assuming smooth solutions. We prove error estimates in L²×L² for the standard Galerkin-FE semidiscretization for the ibvp with u=0 at the boundary, and with characteristic (absorbing) boundary conditions, when the flow is supercritical or subcritical. We test the ability of the numerical method to approximate steady state solutions and “still water” solutions when the system is written in balance-law form. In the final chapter we examine the Discontinuous Galerkin-FE method (DG) for the (SW) system in balance-law form. After an overview of RKDG methods applied to hyperbolic conservation laws in one spatial dimension, we examine slope-limiting procedures. For the RKDG method for the (SW) with variable bottom we consider various issues and algorithms regarding e.g. the well- balancing of the method, the preservation of non-negative water height in case the bottom approach the free surface, and slope limiting procedures in the presence of discontinuities. Finally, we perform a series of numerical experiments of test cases and demonstrate that our algorithms approximate them accurately

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress