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. In addition, we provide detailed descriptions for implementing the proposed methods to run on point clouds. After verifying the convergence rates against the theory, we show that the proposed method is a robust building block for more complicated problems, such as advection problems with non-solenoidal velocity field, inviscid Burgers' equations and systems of reaction advection diffusion equations for pattern formation.

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. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulk-surface coupling are provided for a variety of complex point cloud surfaces.

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. In this paper, we model the interior of a collapsing noble gas bubble as a hard sphere gas driven by a spherical piston boundary moving according to the Rayleigh-Plesset equation. We also include a simplified treatment of ionization effects in the gas at high temperatures. The effects of water vapor are neglected in the model. By using fast, tree-based algorithms, we can exactly follow the dynamics of 10(6) particle systems during the collapse. Our preliminary model shows strong energy focusing within the bubble, including the formation of shocks, strong ionization, and temperatures in the range of 50 000-500 000 K. Our calculations show that the gas-liquid boundary interaction has a strong effect on the internal gas dynamics, and that the gas passes through states where the mean free path is greater than the characteristic distance over which the temperature varies. We also estimate the duration of the light pulse from our model, which predicts that it scales linearly with the ambient bubble radius. As the number of particles in a physical sonoluminescing bubble is within the foreseeable capability of molecular dynamics simulations, we also propose that fine scale sonoluminescence experiments can be viewed as excellent test problems for advancing the art of molecular dynamics.