Seminars during Fall 2013

11 December 2013, 13:15 - 15:00, room 734
Ulrik Skre Fjordholm
An introduction to compensated compactness. Part 2/2.
(See below)
27 November 2013, 13:15 - 15:00, room 734
Ulrik Skre Fjordholm
An introduction to compensated compactness. Part 1/2.
Abstract: In this two-part lecture I will give an introduction to compensated compactness. This method was developed by Murat and Tartar (1978-79) to prove convergence of a sequence of approximate solutions to nonlinear partial differential equations. Since then, it has become an indispensable tool in proving convergence of numerical approximations of many PDEs, especially nonlinear conservation laws. As an application of the theory that we develop, I will prove convergence of a viscous regularization of a scalar conservation law, and – if time permits – prove convergence of a numerical method. Most of the results that we will need from analysis and the theory of conservation laws will be repeated, but some familiarity with measure theory and Sobolev spaces will be of help.
Further reading:
- Chapter 3 of Weak and Measure-Valued Solutions to Evolutionary PDEs by J. Necas, J. Malek, M. Rokyta and M. Ruzicka
- Chapter 16 of Hyperbolic Conservation Laws in Continuum Physics by C. Dafermos
13 November 2013, 13:15 - 14:00, room 734
Ulrik Skre Fjordholm
Numerical approximation of measure-valued solutions to hyperbolic conservation laws.
Abstract: One-dimensional hyperbolic systems of conservation laws are well-posed under the assumption that the initial data is "sufficiently small". However, there is no stability, existence or uniqueness theory for general initial data in multiple dimensions, and certain Cauchy problems might indeed be unstable with respect to initial data. We advocate the point of view of so-called measure-valued solutions, and give numerical evidence that this might be the correct notion of solutions for hyperbolic conservation laws. We prove the existence and stability of measure-valued solutions in certain special cases, and design numerical algorithms that show strongly convergent behavior in unstable Cauchy problems.
6 November 2013, 13:15 - 14:00, room 734
Gabriele Villari, Florence, Italy
An improvement of Massera’s theorem for the existence and uniqueness of a periodic solution for the Lienard equation.
Abstract: In this talk we prove the existence and uniqueness of a periodic solution for the Lienard equation x'' + f (x) x' + x = 0. The classical Massera’s monotonicity assumptions, which are required in the whole line, are relaxed to the interval (h,k), where and h and k can be easily determined.
23 October 2013, 13:15 - 14:00, room 734
Evelyn Buckwar, Linz, Austria
Stochastic numerics and issues in the stability analysis of numerical methods.
Abstract: Stochastic Differential Equations (SDEs) have become a standard modelling tool in many areas of science, e.g., from finance to neuroscience. Many numerical methods have been developed in the last decades and analysed for their strong or weak convergence behaviour. In this talk we will provide an overview on current directions in the area of stochastic numerics and report on recent progress in the analysis of stability properties of numerical methods for SDEs, in particular for systems of equations. We are interested in developing classes of test equations that allow insight into the stability behaviour of the methods and in developing approaches to analyse the resulting systems of equations.
16 October 2013, 13:15 - 14:00, room 734
Trygve Karper, Imperial College
A convergent method for the compressible Navier-Stokes equations.
Abstract: In this talk, I will present a new numerical method for the motion of a viscous isentropic gas in three spatial dimensions. First, we will see that this method is both stable and structure preserving. Then, we will demonstrate that the method is weakly convergent. That is, that the numerical approximation converges to a weak solution as discretisation parameters go to zero. The proof of convergence relies on several tools from functional analysis, such as commutator estimates and the div-curl lemma.
9 October 2013, 13:15 - 14:00, room 734
Lingdi Wang, Fudan University, Shanghai
An Energy-Conserving Second Order Numerical Scheme For Nonlinear Hyperbolic Equation With An Exponential Nonlinear Term.
Abstract
2 October 2013, 8:15 - 9:00, PFI Building, Auditorium 5th floor
Tom Huges, Austin, USA
Isogeometric phase-field modelling of brittle and ductile fracture.
2 October 2013, 13:15 - 14:00, room 734
Zhenhua Chen, Fudan University, Shanghai.
Numerical Simulations for Some New Epitaxial Thin Film Models.
Abstract
2014-01-07, ulriksf