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Many physical systems such as fluid dynamics are governed by a coupled nonlinear evolutionary system of equations. In order to effectively compute numerically such a coupled system of nonlinear PDEs, it is necessary to linearize and decouple the equations at a certain stage. For example, explicit treatment of nonlinear and global terms in time discretizations is a practical and effective way of decoupling complicated systems. On the other hand, some of the linear operators such as the diffusion terms ought to be treated implicitly due to efficiency considerations. It is the proper combination of explicit and implicit treatment of various linear and nonlinear terms that results in an accurate and efficient numerical scheme. The art of time discretization, is therefore to find a suitable operator decomposition and estimate the spectrum of each operator in order to insure the stability of the scheme under reasonable time step size. In this talk numerical schemes are presented for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the convection and pressure terms are treated explicitly in time, with the incompressibility constraint replaced by a pressure Poisson equation. The computation of the momentum and kinematic equations are fully decoupled, resulting is a class of extremely efficient Navier-Stokes solvers. The cost of solving 3D Navier-Stokes equation is therefore comparable to solving a heat equation and a Laplace equation. These methods work well for both large and small Reynolds number flows. Moreover, the schemes are not projection-type methods and are free of numerical boundary layers resulting from time consistency issues inherent in such splitting methods. Full time accuracy is achieved for all flow variables in the L^\infty norm. In addition to efficiency, this class of schemes enjoys some remarkable stability properties. Indeed, a first order semi-implicit discretization of these explicit pressure treatment schemes has been proven to be unconditionally stable. Additionally, since the key to the schemes is the proper pressure Poisson formulation at the PDE level, any kind of spatial discretization such as finite difference, spectral (collocation and Galerkin) methods, and finite element method can be incorporated into the scheme. In particularly, standard continuous finite element spaces can be used to handle the general 3D domains. Moreover, it is proven that the so called inf-sup compatibility condition need not be imposed on the pressure and velocity spaces if the elements are at least $C^1$.
Gradient reaction-diffusion systems arise in the context of modeling the kinetics of second-order or weakly first-order phase transitions, with a broad range of applications. These systems are known to exhibit a variety of non-trivial spatio-temporal behaviors, most notably the phenomenon of propagation and traveling waves. We introduce a variational formulation for the traveling wave solutions in cylindrical geometries, which allows us to construct a certain class of special traveling wave solutions and study a number of their properties. These solutions are special in a sense that they are characterized by a non-generic fast exponential decay ahead of the wave and play an important role in propagation phenomena for the initial value problem. In particular, we show that no solution of the initial value problem that is initially sufficiently localized can propagate faster than the speed of the obtained traveling wave. We also show that only this type of traveling wave solutions can be selected as the asymptotic limit of the solution in the reference frame associated with its leading edge at long times. The considered variational formulation gives easily verifiable upper and lower bounds for propagation speeds. This is joint work with M. Lucia and M. Novaga.
We want to provide rigorous results for experimental observations of vortices in Bose-Einstein condensates: when the trap holding the atoms is rotated, vortices are observed in the system: for intermediate velocity, there is a single bending vortex line, while for large velocity, the number of vortices increases and a lattice is formed. We investigate the behavior of the wave function minimizing the Gross Pitaevskii energy. We find the critical velocity for the nucleation of the first vortex and provide a justification for the structure of the vortex line, which relies on the analysis of a reduced energy depending on the vortex line only. For a fast rotating condensate, we investigate the structure of the lattice using wave functions in the Lowest Landau Level. We find that the minimizer has a distorted vortex lattice, determine the optimal distortion and relate it to the decay of the wave function.
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We consider evolution equations which are invariant under a group of scaling transformations. Our main examples are the nonlinear heat and Schrödinger equations, as well as the Navier-Stokes system. We show that there exist solutions which are asymptotic to different self-similar solutions along different time sequences going to infinity. In fact, the set of self-similar solutions obtainable in this fashion as an asymptotic limit from a single solution can be infinite dimensional. Furthermore, the flow operator at a fixed positive time induced by the equation, followed by an appropriate spatial dilation, generates a chaotic discrete dynamical system. This is joint work with Thierry Cazenave and Flavio Dickstein.
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For questions, contact Eduard Kirr at email@example.com