Unit:
Department of MathematicsLibrary of the School of Science
Author:
Kounadis Grigorios
Dissertation committee:
Βασίλειος Δουγαλής, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Μιχαήλ Δρακόπουλος, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Δημήτριος Θηλυκός, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Θεόδωρος Κατσαούνης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Κρήτης
Μαριλένα Μητρούλη, Καθηγήτρια, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Σωτήριος Νοτάρης, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Ιωάννης Στρατής, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Original Title:
Numerical methods for Shallow Water Equations
Translated title:
Numerical methods for Shallow Water Equations
Summary:
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.
Main subject category:
Science
Keywords:
Shallow water equations, Boussinesq systems, numerical methods, Standard and Discontinuous Galerkin finite element method, error estimates, characteristic boundary conditions, surface dispersive long-wave propagation, solitary waves, variable bottom topography
