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PDE/Applied Math Seminar - Abstracts Fall 2003

(Oct. 2) Jack Xin PDE Modeling of the Ear and Sound Signal Processing

We discuss a class of dispersive nonlinear PDEs based on mechanics and neural functions of the inner ear (cochlea), then show analytical and numerical properties of wave like solutions under single and multiple frequency tonal inputs. A resulting nonlinear phenomenon is masking due to interaction of tones or tone and banded noise. We present a first principle based two level psychoacoustic model of masking and relate it to sound compression in digital music (MP3).

(Nov. 6) Seung-Yeal Ha Lyapunov functionals for the Enskog-Boltzmann equation

In my talk, I will present two Lyapunov functionals measuring potential interactions between particles with different velocities, and L^1-distance between two classical solutions for the Enskog-Boltzmann equation when interactions between particles with different velocities, and L^1-distance between two classical solutions for the Enskog-Boltzmann equation when initial datum is a small perturbation of a vacuum and tends to zero fast enough at infinity in the phase space. These Lyapunov functionals yield time-asymptotic convergence of classical solutions to the collision free motion and the L^1 stability of classical solutions.

(Nov. 13) Tom Hou Efficient Numerical Computations of Stochastic Partial Differential Equations

Stochastic PDE's with solutions depending on multiple scales play fundamental and important roles in many problems. Numerical simulations become an important strategy in practical applications. Monte Carlo methods are used widely, but convergence is usually slow and accuracy is poor. Here we propose a strategy, which uses Wiener Chaos expansions, to design efficient numerical methods for stochastic PDE's. One of the main advantages of this approach is that it separates randomness from the problems. We can calculate all statistical properties of the solutions by solving the induced deterministic coefficient equations. This approach does not use random number generating in the computations. Thus, well developed numerical techniques can be used to solve the coefficient equations. We apply this approach to solve the randomly forced Bousinesq equations and Burgers' equation. We demonstrate that our approach is more efficient than the traditional Monte-Carlo method in achieving a comparable accuracy. Moreover, the method gives better control over the computational errors. Finally, we have performed careful long time computations for the randomly forced Bousinesq equations to study the mixing property of the unstably stratefied fluid flow.

(Nov. 14) Tom Hou Multiscale Modeling and Computation of Flow in Heterogeneous Media

Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. In this talk, I will give an overview of the multiscale finite element method and describe some of its applications, including composite materials, wave propagation in random media, convection enhanced diffusion, flow and transport in heterogeneous porous media. It is important to point out that the multiscale finite element method is designed for problems with many or continuous spectrum of scales without scale separation. Further, we introduce a new multiscale analysis for convection dominated 3-D incompressible flow with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions locally in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. Our multiscale analysis provides an important guideline in designing a systematic multiscale method for computing incompressible flow with multiscale solutions.

(Nov. 20) Robin Young The vacuum in isentropic gas dynamics

We consider the equations of isentropic gas dynamics with large data. A difficulty in analysing this system is the possible appearance of a vacuum, near which Glimm's famous wave interaction estimates break down. I shall describe why this happens and what can be done about it, and present an approximation scheme for modelling solutions which include vacuums. I'll also discuss the can be done about it, and present an approximation scheme for modelling solutions which include vacuums. I'll also discuss the structure of solutions containing vacuums and describe what still needs to be done.

(Dec. 3) Weizhu Bao Numerical Simulation for Bose-Einstein condensation

In this talk, we present numerical methods to compute ground states and dynamics of Bose-Einstein condensates (BEC). As preparatory steps, we take the 3d Gross-Pitaevskii equation (GPE), scale it to obtain a three-parameter model and use an approach well known in the physical literature to reduce it to 2d and 1d GPEs in certain limiting regimes. Two numerical methods are presented to compute the ground and excited states and a time-splitting spectral method is used to solve the time-dependent GPE for dynamics. Then the numerical methods are applied to study collapse and explosion of BEC, as well as stability and interaction of central vortex states in BEC. Comparison of our numerical results and experiment data are also presented.

(Dec. 4) Omar Lakkis Finite Element Computation of Epitaxial Growth with Attachment-Detachment Kinetics

We introduce an adaptive FEM (finite element method) for a free boundary problem modeling island dynamics in epitaxial growth. The physical situation is modelled mathematically by:

- an adatom (adsorbed atom) diffusion equation on terraces of different height;

- boundary conditions on each terrace boundary that include the kinetic asymmetry in the adatom attachment and detachment from each side of the boundary;

- the motion of the terrace boundaries according to a law such that the normal velocity is determined by a two-sided flux and a one-dimensional ``surface'' diffusion.

We solve the problem using two independent triangulations: a triangulation of the 2D domain for the adatom diffusion and a partition in segments for the boundary evolution. The diffusion equation is discretized by a first-order implicit scheme in time and the linear finite element method in space for which a technique of "extension" is used in order to avoid the complexity in the spatial discretization near boundaries. The terrace height (an integer) is tracked by assigning to (marking) each simplex of the triangulation; the marking is updated in each time step. The evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements. Practical adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem.

We show numerical tests for the pure geometrical motion, the mass balance and the stability of a growing circular island. These tests, compared with exact solutions, demonstrate that the method is stable, efficient, and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period. Numerical tests for more complicated situations are also exhibited to illustrate the applicability of the method to real life situations.

Joint work with: E. Baensch (WIAS, Berlin), F. Hausser (caesar, Bonn), B. Li (UMD College Park, Maryland), A. Voigt (caesar, Bonn).

Georg Dolzmann

Last modified: Tue Dec 2 14:01:18 EST 2003