*Please note, this talk will be held virtually via WebEx*

Nonlinear Acoustics: Fundamental Concepts and Shock Applications

I will review the fundamental concepts and equations of nonlinear acoustics theory, derive equations of motion in terms of the scalar velocity potential, and present a number of example problems involving shock phenomena.

February 20

Fredrik Fryklund, NYU

An Integral Equation Method for the Advection-Diffusion Equation on Time-Dependent Domains in the Plan

Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complicated geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

March 6

Pablo Ravazzoli, The Center for Research in Physics and Engineering of the Center of the Province of Buenos Aires (CIFICEN)

Equilibrium Solutions of 3 and 4 Phase Systems

We study the equilibrium shape of liquid lens (two liquids and a gas phase) and liquid bridges (two liquids, gas and solid phase) considering the surface tension forces in nonwetting situations (negative
spreading factor). For the lenses, we obtain analytical expressions for the drop shape when gravity can be neglected and when gravity is include in the analysis, we find two different families of equilibrium solutions for the same set of
physical parameters. For the bridges, we did not consider the gravity effects, but we also found more than one solution for the same set of physical parameters. In both cases we determine the family that has the smallest energy, and therefore is the most probable to be found in nature.

April 3

Zewen Shen, University of Toronto

Is Polynomial Interpolation in the Monomial Basis Unstable?

In this talk, we will show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. We will also show that the monomial basis is more advantageous than other polynomial bases in a number of applications.