Spring 2016
| Time/place: Thursday 23 June at 14:15–15:00 in room 656 |
|---|
| Speaker: Gabriele Brüll Title: On the symmetry of traveling wave solutions to the Whitham equation |
| Abstract: The Whitham equation is a nonlocal, nonlinear dispersive wave equation introduced by G. B. Whitham as an alternative wave model equation to the Korteweg–de Vries equation, describing the wave motion at the surface on shallow water. Knowing that traveling wave solutions to the Whitham equation exist, we prove that any solitary wave solution is symmetric and has exactly one crest. Moreover, the structure of the Whitham equation allows to conclude that conversely any classical symmetric solution constitutes a traveling wave. In fact, the latter result holds true for a large class of partial differential equations sharing a certain structure. |
| Time/place: Thursday 30 June at 14:15–15:00 in room 656 |
| Speaker: Lars Hov Odsæter Title: Projection of flux for local conservation |
| Abstract: Conservation of mass is of great importance for fluid problems. Not all numerical schemes produce fluxes that are locally conservative. In this talk we present a postprocessing method to project any non-conservative flux to a subspace that is locally conservative. This is done by adding a piecewise constant correction term that is minimized in a weighted \(L^2\) norm. We formulate this as a variational problem and prove existence and uniqueness. Furthermore, we prove that the postprocessed flux has the same order of convergence as the original flux approximation. Finally, we will demonstrate our method by some numerical examples and couple the flux approximation to an advective transport problem to illustrate the importance of local conservation. |
| Time/place: Thursday 9 June at 14:15–15:00 in room 656 |
| Speaker: Anders Samuelsen Nordli Title: \(\alpha\)-dissipative solutions of the Hunter–Saxton system |
| Abstract: Solutions of the Hunter–Saxton system experience wave-breaking in finite time. Past wave-breaking there are several ways to continue weak solutions, the ones most studied being conservative and dissipative solutions. Here we consider a novel solution concept, namely \(\alpha\)-dissipative solutions, that dissipates an \(\alpha\)-part of the energy at wave-breaking. Moreover, we let \(\alpha\) vary across space, which makes it difficult to predict what value \(\alpha\) will take at wave-breaking. We look at existence of solutions and continuous dependence on initial data. Some examples will be discussed. This work in progress is a joint work with Katrin Grunert. |
| Time/place: Thursday 2 June at 14:15–15:00 in room 656 |
| Speaker: Charles Curry Title: Lie group integrators and coordinates of the second kind |
| Abstract: We discuss the problem of geometric integration and in particular Lie group integration. Special attention is given to the role of coordinate maps, notably coordinates of the second kind. Here the coordinate mapping depends on a choice of ordered basis for a given Lie algebra. The performance of integration schemes employing these coordinates may depend heavily on the choice of basis. In a refinement of results of Owren and Marthinsen, we show how the theory of root systems gives rise to bases of complex semisimple Lie algebras with desirable properties. We discuss extensions of these methods to more general Lie algebras. |
| Time/place: Thursday 26 May at 14:15–15:00 in room 656 |
| Speaker: Filippo Remonato Title: Two-dimensional bifurcation in the Whitham Equation with surface tension |
| Abstract: The Whitham equation arises in the modelling of free-surface waves in irrotational, incompressible fluids (water), and combines a generic nonlinear quadratic term with the exact linear dispersion given by the Euler's equations. The use of the exact dispersion relation allows for a better representation of the propagation of shorter waves than, for example, the KdV equation, but also poses some mathematical challenges. The classical model deals with purely gravitational waves, and in this setting several results have recently been produced on the existence and qualitative properties of the waves. In this work, we follow a natural extension and focus our attention on gravity-capillary waves, incorporating the effects of surface tension in the phase speed. We show the existence of both bifurcation curves and bifurcation sheets of nontrivial solutions, as well as the existence of multimodal waves. Numerical computation will allow us to investigate the branches and visualise the wave profiles, and some illustrative examples will be presented. |
| Time/place: Thursday 19 May at 14:15–15:00 in room 656 |
| Speaker: Atanas Stefanov, University of Kansas Title: Travelling waves for granular chains |
| Abstract: In this talk, we consider a model of a (double infinite) granular chain interacting only with the nearest neighbour through a Hertzian force (physically, one should think about an infinite line of tightly packed metal beads). This is a highly non-linear model, for which we show that travelling waves exist. In addition, we show such waves are bell-shaped and they are compactons (i.e. they have an unusual double exponential rate of decay). We also consider the same issues for more sophisticated models (with pre-compressed beads), as well as the mass-in-mass system (in which each bead has an internal mass embedded into it). Our approach is through calculus of variations for the corresponding non-standard functionals, but some simple Fourier analysis comes into play as well. This is based on a series of papers with Panos Kevrekidis (University of Massachusetts). |
| Time/place: Friday 20th of May at 13:15–14:00 in room 734 |
| Speaker: Toshiaki Itoh (Doshisha University) Title: Discretization of ODEs using Symmetries |
| Abstract: The topic of this talk is the algebraic construction of the exact numerical integrator for ODEs. This approach can sometimes be used also to obtain discrete integrable system of ODEs. By exploring the varieties of exact discretization of ODEs which are constructed from their symmetries with symbolic calculations, we obtain hints about how to construct new numerical integrators for ODEs and about how to find their solutions. |
| Time/place: Thursday 12 May at 14:15–15:00 in room 656 |
| Speaker: Susanne Solem Title: Second-order convergence of finite volume schemes for conservation laws |
| Abstract: Although the development of numerical methods for conservation laws has been a very active field for several years, there is still a large gap between observed accuracy and theoretically provable convergence rates for these methods. To fill this gap, it seems promising to investigate convergence rates in metrics which are more suitable to the problem at hand than e.g. \(L^1(\mathbb{R})\). The Wasserstein distance, a metric on the set of probability measures, seems to be very suitable for scalar conservation laws, both heuristically, theoretically and in practical computations. A result supporting these observations will be presented. In particular, we will show that whenever the initial data \(u_0\) is decreasing and consists of a finite number of piecewise constants, a class of monotone schemes for the scalar conservation law \(u_t +f(u)_x=0\), with \(u(x,0)=u_0(x)\) and \(f\) convex, converges to the exact entropy solution at a rate of \(\Delta x^2\) in the Wasserstein distance. |
| Time/place: Thursday 10 March at 14:15–15:00 in room 656 |
| Speaker: Henrik Kalisch (University of Bergen) Title: Nonlinear models for wave shoaling and wave breaking |
| Abstract: Waves transport mass, momentum and energy. In shoaling processes, wave energy is generally conserved, while wave momentum may vary as the waves propagate towards the beach. The linear theory of wave shoaling utilizes energy conservation to obtain changes in waveheight as a function of the local undisturbed depth. The momentum balance is usually expressed in terms of the radiation stress tensor, and it can be shown using the radiation stress that momentum transport in a shoaling wavetrain varies with decreasing depth, leading to local changes in the mean water level. The change in the mean water level is called setdown. We will review the classical linear theory of shoaling, and then explain some attempts to extend this theory to the nonlinear case. For the nonlinear case, momentum and energy balances in the context of the KdV equation are used in conjunction with periodic cnoidal wave solutions. The KdV Equation also features convective wave breaking, and we will explain how a kinematic criterion can be used to understand incipient wave breaking in the context of the KdV equation. |
| Time/place: Thursday 3 March at 14:15–15:00 in room 656 |
| Speaker: Yuexun Wang (Tsinghua University, Beijing) Title: A class of degenerate equation with applications in fluid mechanics |
| Abstract: In this talk, we will introduce some recent progresses on the non-existence in Sobolev space to the Cauchy Problem of the compressible Navier-Stokes equations and Schauder estimates for one class of parabolic operator degenerate both on time and space. |