Meshfree Semi-Lagrangian Methods for Solving Surface Advection PDEs.

We analyze a class of meshfree semi-Lagrangian methods for solving advection problems on smooth, closed surfaces with solenoidal velocity field. In particular, we prove the existence of an embedding equation whose corresponding semi-Lagrangian methods yield the ones in the literature for solving problems on surfaces. Our analysis allows us to apply standard bulk domain convergence theories to the surface counterparts.

Simple computation of reaction-diffusion processes on point clouds.

The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators.

Molecular dynamics of extreme mass segregation in a rapidly collapsing bubble.

A molecular dynamic simulation of a mixture of light and heavy gases in a rapidly imploding sphere exhibits virtually complete segregation. The lighter gas collects at the focus of the sphere and reaches a temperature that is several orders of magnitude higher than when its concentration is 100%. Implosion parameters are chosen via a theoretical fit to an observed sonoluminescing bubble with an extreme expansion ratio (25:1) of maximum to ambient radii.

Molecular dynamics simulation of the response of a gas to a spherical piston: implications for sonoluminescence.

Sonoluminescence is the phenomena of light emission from a collapsing gas bubble in a liquid. Theoretical explanations of this extreme energy focusing are controversial and difficult to validate experimentally. We propose to use molecular dynamics simulations of the collapsing gas bubble to clarify the energy focusing mechanism, and determine physical parameters that restrict theories of the light emitting mechanism.