DEPARTMENT OF MATHEMATICS AND STATISTICS

09/12 Catalin Turc, Mathematics, NJIT , Efficient solutions of periodic scattering problems.

We present an accurate and efficient numerical method, based on integral Nystrom discretizations, for the solution of three dimensional wave propagation problems in piece-wise homogeneous media that have two-dimensional (in-plane) periodicity (e.g. photonic crystal slabs). Our approach uses (1) A fast, high-order algorithm for evaluation of singular integral operators on surfaces in three-dimensional space, and ( 2) A new, representation of the three-dimensional quasi-periodic Green's functions, which, based on use of infinitely-smooth windowing functions, and equivalent-source representations, converges super-algebraically fast throughout the frequency spectrum. Our fast algorithm for computing periodic Green's functions compare favorably with the classical Ewald's summation method and with other existing methods for fast summations of periodic Green's functions.

10/10, Richard McLaughlin, UNC Chapel Hill, Lagrangian Tori induced by precessing bent rods in viscous dominated flows

Motile cilia play a large role in fluid motion across the surface of ciliated tissue. We present an experimental and theoretical study involving a single rigid cilium rotating in a viscous fluid about one of its ends in contact with a horizontal no-slip plane. Experimentally tracked three dimensional Lagrangian trajectories are compared with theoretical trajectories computed using a properly imaged slender body theory. The addition of planar bend to the rod geometry is shown to break symmetry and create large scale nested tori in the Lagrangian particle trajectories. Three dimensional PIV measurements are presented which help to explain the origin of the large scale tori and compared directly with the slender body theory.

10/24, Weiqing Ren, Singapore University, The string method for the study of rare events

Many problems arising from applied sciences can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well known examples include nucleation events during phase transitions, conformational changes of bio-molecules, chemical reactions, etc. The dynamics proceeds by long waiting periods in metastable states followed by sudden jumps or transitions from one metastable state to another. In this talk, I will discuss the string method for the study of complex energy landscapes and the extension of the string method for saddle points search.

11/21, Thanksgiving

11/28, Matthew Williams, PACM, Princeton, Exploiting Low Dimensionality in Nonlinear Optics and Other Physical Systems

Due to their prevalence in physical applications, the dynamics and behavior of nonlinear PDEs and large coupled systems of nonlinear ODEs are of interest to scientists and engineers; mathematically, we desire the solution branches and bifurcations of the underlying system. However, the computational cost associated with using standard methods for numerical continuation can often be prohibitive due to the large state space. In this talk, I will discuss the application of data driven methods such as the Proper Orthogonal Decomposition to the Waveguide Array Mode-Locking (WGAML) Model in nonlinear optics and Eulerâ€™s equations for standing surface water waves in order to generate reduced order models (ROMs). Using the resulting ROMs, I then compute the solutions and bifurcations involved in the multi-pulsing transition of WGAML model and the solution branches of time-periodic standing surface waves. I will also discuss the generalization of these data-driven methods for other uses, such as in an adaptive ROM/PDE integrator.

12/5, Victor Dominguez, Universidad Publica de Navarra, Spain, Robust methods for highly oscillatory integrals

Numerical methods for approximating very oscillatory integrals have gained a renewed interest in the last years. For instance, in many numerical methods for solving scattering problems in the high frequency domain, there appear integrals whose oscillations are proportional, at best, to the wave number of the incident wave. The very well-known rule of thumb of using 10-20 point per oscillation to capture the integrals properly leads to really prohibitively expensive methods, even in the simpler problems. Furthermore the integrand itself can be singular which has to be considered too. In this talk we present a Filon-Clenshaw-Curtis approach for approximating such integrals. The analysis we present for these rules is complete: We deduce convergence rates, in terms of the number of nodes and the "strength" of the oscillations, we show an efficient way to implement the resulting algorithms, and we prove its numerical stability. The use of graded meshes, needed to take care of any singularity in the integrand, is also explored. We show that, with the right choice of the grading parameters, the original good convergence rates can be restored in this case too. This is a joint work with Tatiana Kim and Ivan G. Graham (University of Bath, UK) and V.P. Smyshlyaev (University College London, UK).

12/12, Bill Schultz, University of Michigan, Oscillating contact lines

ODE free surface boundary conditions become PDEs for non-zero surface tension when applied to potential flow, hence the problem of small surface tension is singular. Even for water in moderately sized containers, most dissipation of wave energy occurs in the vicinity of the contact line (where the interface meets the solid walls) and where the point BCs are placed on the Line BC. The standard contact line models designed for low Reynolds number and unidirectional motion are inadequate--in fact they provide no dissipation. We borrow techniques from viscoelastic flow to put history effects into the contact line model and compare to measurements of slugs of water inside glass tubes. We then obtain realistic dissipation--even with potential flow in the interior.

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