arXiv subjects

Stanly L. Steinberg

Publications and source records attributed to Stanly L. Steinberg.

2 recordsLinked to original sources

arXiv2019

Mimetic Explicit Time Discretiztions

This paper is part of a program to combine a staggered time and staggered spatial discretization of continuum mechanics problems so that any property of the continuum that is proved using vector calculus can be proven in an analogous way for the discretized system. We require that the discretizations be second order accurate and have a conserved quantity that approximates the energy for the system and guarantees stability for a reasonable constraint on the time step. We also require that the discretization is time explicit so as to avoid the solution of large system of possibly nonlinear algebraic equations. The well known Yee grid discretization of Maxwell's equations is the same as our discretization and is an early example of using a staggered space and time grid . To motivate our discussion we begin by studying the staggered time or leapfrog discretization of the harmonic oscillator and use this to introduce the modification of the energy that is conserved. Next we use systems of linear equations to motivate the definition of the modified energy for more complex systems of ordinary differential equations and then apply our ideas to the scalar wave equation in one spatial dimension. We finish by discretizing the three dimensional scalar wave and Maxwell's equations. Because the spatial discretization is mimetic, we obtain that the divergence of the electric and magnetic fields are constant when there are no sources. Using the mimetic properties the proof of this trivial and is essentially the same as in the continuum.

math.NA↗

arXiv2015

A New Characterization of Fine Scale Diffusion on the Cell Membrane

We use a large single particle tracking data set to analyze the short time and small spatial scale motion of quantum dots labeling proteins in cell membranes. Our analysis focuses on the jumps which are the changes in the position of the quantum dots between frames in a movie of their motion. Previously we have shown that the directions of the jumps are uniformly distributed and the jump lengths can be characterized by a double power law distribution. Here we show that the jumps over a small number of time steps can be described by scalings of a {\em single} double power law distribution. This provides additional strong evidence that the double power law provides an accurate description of the fine scale motion. This more extensive analysis provides strong evidence that the double power law is a novel stable distribution for the motion. This analysis provides strong evidence that an earlier result that the motion can be modeled as diffusion in a space of fractional dimension roughly 3/2 is correct. The form of the power law distribution quantifies the excess of short jumps in the data and provides an accurate characterization of the fine scale diffusion and, in fact, this distribution gives an accurate description of the jump lengths up to a few hundred nanometers. Our results complement of the usual mean squared displacement analysis used to study diffusion at larger scales where the proteins are more likely to strongly interact with larger membrane structures.

q-bio.QM↗