The DNA Seminar

The Differential Equations and Numerical Analysis Seminar

This seminar is a continuation of two seminar series: One on numerical analysis and one on differential equations (DIFTA).

Seminars of the spring semester 2012

Date	Speaker and title
13.06.12 14:15 - 15:00 room 734	Knut Waagan (University of Washington) Numerics for ideal MHD in 1,2 and 3 space dimension. I will present positive and entropy stable schemes for the system ideal MHD equations. This is a nonstrictly hyperbolic system of conservation laws constituting a fluid model for plasmas. It is an important model in astrophysics. The schemes are finite volume schemes based on approximate Riemann solvers. When generalizing to multi-dimensions, special care needs to be taken due to the structure imposed by the magnetic field having zero divergence. The scheme has, due to its robustness enabled simulation of realistic flow regimes in the interstellar medium.
05.06.12 12.15-13:00 room 734	Long Pei (Zhejiang University) Existence and uniqueness of generalized monopoles in six-dimensional non-Abelian gauge theory Abstract: We established the existence and uniqueness of the spherically symmetric monopole solutions in SO(5) gauge theory with Higgs scalar field- s in the vector representation in six-dimensional Minkowski space-time and obtain sharp asymptotic estimates for the solutions. Our method is based on a dynamical shooting approach that depends on two shooting parameters which provides an effective framework for constructing the generalized monopoles in six-dimensional Minkowski space-time.
03.05.12 14:15-15:00 room 734	Rolf Jeltsch (ETH) Numerical Simulation of Compressible Magnetohydrodynamic Plasma Flow in a Circuit Breaker Rolf Jeltsch, Ralf Hiptmair, Patrick Hugueniot, Harish Kumar, Christoph Schwab, Manuel Torrilhon, Vincent Wheatley The main function of a circuit breaker is to switch off the electric current safely, in case of fault current. A mechanical force separates the contacts, and an arc starts to burn between the two contacts. This plasma is described by the resistive Magnetohydrodynamics (MHD) equations. The emphasis is on very high currents (10kA-200kA) and relatively high conductivity. Radiation is incorporated by adding a Stefan's radiation. To simulate the plasma in the arc the Nektar code developed by Brown University is adapted and extended. It is based on the Discontinuous Galerkin(DG) methods allowing for triangular or quadrilateral meshes in 2d and hexagonal or tetrahedral meshes in 3d. GID is used for mesh generation. The code is extended to include Runge-Kutta time stepping, various accurate Riemann solvers for MHD, slope limiters and $SF_6$ gas data. It operates on both serial and parallel computers with arbitrary number of processors.The suitability of this Runge-Kutta Discontinuous Galerkin (RKDG) methods is analysed. In particular different numerical fluxes, different Riemann solvers and limiters, low and high order approximations on smooth and non-smooth solutions are investigated. Numerical results are given. This work has been performed by Patrick Huguenot and Harish Kumar in their Ph.D. thesis and by Vincent Wheatley.
02.05.12 14:15-15:00 room 734	Daniel DaSilva (University of Rochester, NY) Global regularity in generalized wave maps Wave maps are nonlinear generalizations of the wave equation which have been studied for decades. In this talk, we will consider generalizations of wave maps based on the Skyrme and Adkins-Nappi models of nuclear physics. These models yield nonlinear hyperbolic partial differential equations, for which we consider the question of regularity of solutions. In particular, we will discuss the non-concentration of energy in these models, a preliminary step in establishing a global regularity theory.
25.04.12 14:15-15:15 Rm 734, S2	Anton Evgrafov, Department of Mathematics, DTU, Denmark Optimization approaches to rational geometric design of engineering systems, Many important problems encountered in industrial design amount to discovering a geometric shape, which satisfies the desired functional and manufacturing requirements. Most often it is also desirable that the shape maximizes a prescribed performance functional. Classical examples of such problems include finding the lightest elastic solids of given stiffness and strength, or finding aerodynamic shapes of a given volume with prescribed lift and drag characteristics. Mathematically, these problems translate into finding an optimal domain for a system of PDEs, governing the engineering system under consideration. We will discuss two alternative computational approaches to this class of problems, based on control in the coefficients of PDEs (a.k.a. topology optimization) and on isogeometric analysis.
26.03.12 12:15-13:00 room 734 SB2	Klas Modin Higher dimensional generalisation of the µ–Hunter–Saxton equation. A higher dimensional generalisation of the µ–Hunter–Saxton equation is presented. This equation is the Euler-Arnold equation corresponding to geodesics in Diff(M) with respect to a right invariant metric. It is the first example of a right invariant non-degenerate metric on Diff(M) that descends properly to the space of densities Dens(M) = Diff(M)/Diffvol(M). Some properties and results related to this equation are discussed. In particular, a result related to the polar decomposition theorem for diffeomorphisms.
20.02.12 12:15-13:00 room 734 SB2	Olivier Verdier Geometric Rattle. Constrained mechanical systems (robots, rod models) have to be simulated with care. In particular, it is important to design numerical integrators which preserve the "mechanical structure" of the system. Those integrators are known, for instance, to approximately preserve energy and other invariants. I will give a geometric description of the existing structure preserving integrators for constrained mechanical systems (called "Shake" and "Rattle"). Finally I will explain how to extend those methods to handle cases that were out of reach for the current solvers. (Joint work with K. Modin and R.I. McLachlan)
13.02.12 12:15-13:00 room 734 SB2	Sergio Vessella (Firenze) Three sphere inequalities for the anisotropic plate equation and applications. We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coeffcients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. (Joint work with A. Morassi, E. Rosset)
23.01.12 12:15-13:00 room 734 SB2	Mahmoudreza Bazarganzadeh (KTH) Introduction to two phase quadrature domains: Theory and numerical solutions Two phase quadrature domains (QD) was introduced in 2009 and the existence was proved by considering some restrictions. I will introduce this new topic with examples and discuss a little bit on uniqueness. Then I will turn to numerical methods based on finite difference method to find two phase QD. We construct monotone finite difference scheme which converges to viscosity solution of given problem. We mainly use Delta dirac function and will implement the finite difference schemes to find appropriate approximation for this problem.
11.01.2012 14:15-15:00 room 1329	Trygve Karper (Maryland) On some biological models of self-organization In nature, groups of individuals organize globally using only local information. For instance, in a school of fish, there are no external forces to coordinate the group, no leader to guide them. Several mathematical models have been proposed to describe self-organization. In this talk, we will consider two of the most famous models; the Keller-Segel model for bacterial chemotaxis and the Cucker-Smale model for flocking. The emphasis will be on recent mathematical results and remaining open questions. Starting from the microscopic description on particle level, we will discuss the derivation of mesoscopic and macroscopic models at the continuum level. Since the motivation for deriving continuum models is largely numerical, we will also discuss distinct numerical challenges with the resulting models.