The DNA Seminar
The Differential Equations and Numerical Analysis Seminar
Earlier this semester:
| Thursday 4 December, 13:15-14:00 in room 734 |
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| Erik Wahlén (Lunds Tekniska Högskola) Title: Solitary water waves in three dimensions |
| Abstract: I will discuss some existence results for solitary waves with surface tension on a three-dimensional layer of water of finite depth. The waves are fully localized in the sense that they converge to the undisturbed state of the water in every horizontal direction. The existence proofs are of variational nature and different methods are used depending on whether the surface tension is weak or strong. In the case of strong surface tension, the existence proof also gives some information about the stability of the waves. The solutions are to leading order described by the Kadomtsev-Petviashvili I equation (for strong surface tension) or the Davey-Stewartson equation (for weak surface tension). These model equations play an important role in the theory. This is joint work with B. Buffoni, M. Groves and S.-M. Sun. |
| Tuesday 2 December, 14:15-15:00 in F4 in Gamle Fysikk |
| Paolo Baroni (Parma, Uppsala) Title: Regularity results for a class of non-autonomous functionals in a borderline case |
| Abstract: We shall present a basic regularity theory for a class of non-autonomous functionals whose energy density changes its ellipticity and growth properties according to the point. The talk is based on a joint work with M. Colombo and G. Mingione. |
| Tuesday 25 November, 14:15-15:00 in room 734 |
| Félix del Teso (Universidad Autonoma de Madrid) Title: Nonlinear fractional diffusion equations. Numerics and propagation properties |
| Abstract |
| Tuesday 18 November, 14:15-15:00 in room 734 |
| Anna Geyer (Universitat Autònoma de Barcelona) Title: On an equation for surface waves of moderate amplitude in shallow water |
| Abstract: In this talk we will discuss traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogeneous fluids. Our approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an explicit expression for bounded orbits in the phase plane to establish existence of the corresponding periodic and solitary traveling wave solutions. Furthermore, we study the orbital stability of solitary traveling waves using a method proposed by Grillakis et.al. |
| Tuesday 11 November, 14:15-15:00 in room 734 |
| Kristoffer Varholm (NTNU) Title: Steady water waves with point vortices |
| Abstract: The study of steady water-wave solutions to the incompressible Euler equations has a long history, but only recently has much attention been paid to rotational waves (waves with nonzero vorticity). Such waves can exhibit interior stagnation points and regions of closed streamlines. The existence of gravity-capillary waves with compactly supported vorticity was established on infinite depth by J. Shatah, S. Walsh, and C. Zeng last year. We extend parts of this result to finite depth, by constructing small-amplitude solitary waves with a point vortex. As opposed to infinite depth, the properties of these waves depend significantly on the position of the vortex in the fluid domain. Waves with multiple point vortices are also considered. |
| Tuesday 4th November, 14:15-15:00 in room 734 |
| Andreas Asheim (NTNU) Title: Using diffraction formulas to formulate the scattering problem for scatterers with edges |
| Abstract: Numerical methods for time-harmonic scattering are usually based on the PDE formulation (Helmholtz) or boundary integral formulations. Each of these approaches have their pros and cons. Recently we have stumbled upon an entirely new kind of integral formulation for the scattering of time-harmonic waves from scatterers with edges, which is based on analytical formulas for diffraction. This integral formulations, it turns out, has several attractive features when it comes to numerical computations, especially in high frequency regimes. In this talk I give an introduction to some of the different paradigms for scattering computations (PDE, BEM and analytic) and present some background on the new integral formulation.This work is joint work with U. P. Svensson from IET. |
| Tuesday 28 October, 14:15-15:00 in room 734 |
| Charles Curry (NTNU) Title: Algebra and numerical simulation of stochastic differential equations driven by Levy processes. |
| Abstract: Solutions of continuous stochastic differential equations with sufficiently smooth driving coefficients permit an expansion in terms of multiple iterated integrals. There exist a number of numerical integration schemes for such equations making use of the algebraic structure of iterated integrals as studied by Chen and others. In this talk we will discuss possible generalizations of such methods to discontinuous equations driven by Levy processes. Such equations are closely related to the jump stochastic differential equations of Ito. We will examine the extent to which the available solution expansions are amenable to the above methods, before considering the impact of the discontinuities on the algebraic structure of multiple iterated integrals appearing in the solution expansions. We will conclude with the presentation of a new class of numerical integration schemes for discontinuous Levy-driven equations that naturally generalize the aforementioned schemes from the continuous case, and discuss the extent to which they inherit the optimality properties of the analogous schemes. |
| Tuesday 21 October, 14:15-15:00 in room 734 |
| Dietmar Hömberg (NTNU and TU Berlin) Title: Modelling, analysis and simulation of multifrequency induction hardening |
| Abstract: Induction hardening is a modern method for the heat treatment of steel parts. A well directed heating by electromagnetic waves and subsequent quenching of the workpiece increases the hardness of the surface layer. The process is very fast and energy efficient and plays an important role in modern manufacturing facilities in many industrial application areas. Although the original process is quite old, recent years have seen an important progress due to a new technology, which allows to work simultaneously with several frequencies in one induction coil. For the first time this technology allows for the contour close hardening of complicated components such as gears in one induction coil. However, the process control especially the adjustment of the frequency fractions is quite delicate and requires costly experiments. Hence there is a hight demand for simulation and optimal control of multifrequency hardening. In my talk I will present some results of a collaboration between two industrial partners and four scientific partners on this topic funded by the German Ministry of Education and Research. In the first part of my talk, a model for multifrequency induction hardening of steel parts is presented. It consists of a system of partial differential equations including Maxwell's equations and the heat equation. We show that the coupled system admits a unique weak solution. In the second part of the talk I will discuss the numerical approximation of the problem. It turns out to be quite intricate since one has to cope with different time scales for heat diffusion and the Maxwell system. Moreover, owing to the skin effect only the boundary layers of the component are heated by induced eddy currents, hence we also have to consider different spatial scales. We present a numerical algorithm based on adaptive edge-finite elements for the Maxwell system which allows to treat these difficulties. We show some 3D simulations and conclude with results of an experimental validation in an industrial setting. |
| Tuesday 14 October, 14:15-15:00 in room 734 |
| Gabriele Brüll (Leibniz Universität Hannover) Title: Two-Phase Thin Film Equations with Insoluble Surfactant |
| Abstract: Consider two fluids on top of each other with a layer of surfactant on the interface of the upper fluid. Assuming the film to be very thin, we use Lubrication Approximation on the governing equations to derive a system of partial differential equations describing the evolution of the two film heights and the surfactant concentration. Neglecting gravitation and taking surface tension to be the only driving force, the system is of fourth order, degenerate and strongly coupled. The aim of this talk is to introduce the problem and presenting two results. The first one is the local well-posedness of the system, which is done by the method of semigroups. The second will be the existence of global weak solutions. The proof is based on using Galerkin approximations and an energy functional, which provides a priori estimates. |
| Tuesday 9 September, 14:15-15:00 in room 734 |
| Helen Parks (UC San Diego) Title: Variational integrators for interconnected Lagrange-Dirac dynamical systems |
| Abstract: This talk presents recent results on interconnecting multiple Lagrange-Dirac systems at the discrete level using variational integrator techniques. We will discuss the variational integrator construction and its desirable properties as well as some background on Dirac structures and Lagrange-Dirac mechanics before presenting the results. |
| Tuesday 2 September, 14:15-16:00 in room 734 |
| Alexander Schmeding Title: Infinite-dimensional Lie theory and the Butcher group |
| Abstract: A Lie group is a group which is also a manifold such that the group operations are smooth. Many Lie groups which arise naturally as groups of symmetry in physics, differential geometry and dynamical systems are infinite-dimensional. Examples for such infinite-dimensional groups include diffeomorphism groups of manifolds, mapping groups and the Butcher group from numerical analysis. The aim of this talk is twofold: First, we shall give an introduction to the infinite-dimensional Lie theory of locally convex groups. The groups envisaged here are Lie groups modelled on locally convex vector spaces (in the sense of [2]). It is necessary to consider such general objects since many of the examples mentioned above can not be modelled on Banach spaces. We remark that the first part of the talk will be an elementary introduction which will prepare the ground for an extended example. In the second part we will report on new results on the Lie group structure of the Butcher group (see [1]). We construct a (real-analytic) Lie group structure for the Butcher group and describe its Lie theoretic properties. Our results complement the purely algebraic treatment of the Butcher group in numerical analysis. The results presented in the second part are joint work with G. Bogfjellmo (NTNU). [1] Brouder, C.: Trees, renormalization and differential equations. BIT Num. Anal.44:425–438, 2004. [2] Neeb, K.H.: Towards a Lie theory of locally convex groups |
| Tuesday 26 August, 14:15-15:00 in room 734 |
| Boris Buffoni (EPFL, Lausanne) Title: On the stability of steady solutions to the full water-wave problem |
| Abstract: Existence of surface water waves can be proved by direct minimization methods based on dynamical invariants. They also lead to stability in some weak sense. In particular I shall consider irrotational solitary waves propagating along a 1D surface and periodic waves with vorticity (joint work with G. R. Burton). |
Workshop in honour of Syvert P. Nørsett's 70th birthday
On the 25th of August there will be a workshop in honour of Syvert.
ROOM Suhm lecture hall (downtown Trondheim)
Programme
- 9:30-10:20, Arieh Iserles, DAMTP Cambridge
- 10:20-11:10, Gustaf Södelind, Lund
- 11:10-11:30, Coffee break
- 11:30-12:20, Reinout Quispel, La Trobe University, Melbourne
- 12:30-13:30, Lunch
- 13:30-14:20, Hans Munthe-Kaas, University of Bergen
- 14:20-…, Refreshments
The talks will take place in room Suhm-aud
Arieh Iserles
Title: Fast computation of the semiclassical Schrödinger equation
Abstract: The computation of the semiclassical Schrödinger equation presents a number of difficult challenges because of the presence of high oscillation and the need to respect unitarity. Typical strategy involves a spectral method in space and Strang splitting in time, but it is of low accuracy and sensitive to high oscillation. In this talk we sketch an alternative strategy, based on high-order symmetric Zassenhaus splittings, combined with spectral collocation, which preserve unitarity and whose accuracy is immune to high oscillation. These splittings can be implemented with large time steps and allow for an exceedingly affordable computation of underlying exponentials. The talk will be illustrated by the computation of different quantum phenomena.
Gustaf Söderlind.
Title: Stiffness and Oscillations. Local damping and frequency in nonlinear systems.
In the past it has been considered difficult to quantify stiffness. It is even harder, however, to quantify oscillatory behavior in nonlinear systems. Some systems are periodic, some are quasi-periodic. Some are Hamiltonian. Some have invariants, others have limit cycles. In this talk we will attempt to quantify highly oscillatory behavior and explore in terms of matrix theory whether oscillations can be associated with the skew-symmetric part of the local Jacobian, in a way that is analogous to the way in which damping is associated with the Jacobian’s symmetric part.
| Tuesday 19 August, 14:15-15:00 in room 734 |
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| Radim Hosek (University of West Bohemia) Title: From generalization of bistable equation to a graph theory problem |
| Abstract: The bistable equation \(u_t = \varepsilon^2 u_{xx} - F'(u)\) is perhaps the simplest model of phase transition at given critical temperature. The function \(F\) represents free energy and usually takes the form of a double-well potential. Abandoning the physical motivation, we investigate behaviour of the system for potentials of other type. A special choice of \(F\) then leads to a basic graph theory problem, which can be solved using combinatorial methods. |