# Difference between revisions of "Applied/ACMS/absS16"

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Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development. | Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development. | ||

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+ | === Lihui Chai (UCSB) === | ||

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+ | ''Hybrid method for computing periodic Schrodinger equation with band crossings'' | ||

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+ | We propose a hybrid method which combine the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with a periodic potential in the band-crossing case. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band crossing a BDTS method is used to capture the important inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. |

## Revision as of 10:52, 4 February 2016

## Contents

# ACMS Abstracts: Spring 2016

### Stefan Llewellyn Smith (UCSD)

*Hollow vortices*

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. After giving a history of point vortices, we obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend on a single non-dimensional parameter. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

### Tom Solomon (Bucknell)

*Experimental studies of reaction front barriers in laminar flows*

We present studies of the effects of vortex-dominated fluid flows on the motion of reaction fronts produced by the excitable Belousov-Zhabotinsky reaction. The results of these experiments have applications for advection-reaction-diffusion dynamics in a wide range of systems including microfluidic chemical reactors, cellular-scale processes in biological systems, and blooms of phytoplankton in the oceans. To predict the behavior of reaction fronts, we adapt tools used to describe passive mixing.In particular, the concept of an invariant manifold is extended to account for reactive burning. Burning invariant manifolds (BIMs) are predicted as one-way barriers that locally block the motion of reaction fronts. These ideas are tested and illustrated experimentally in a chain of alternating vortices, a spatially-random flow, vortex flows with imposed winds, and a three-dimensional, nested vortex flow. We also discuss the applicability of BIM theory to the motion of bacteria in fluid flows.

### Daniele Cappelletti (KU)

*Deterministic and stochastic reaction networks*

Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.

### Lihui Chai (UCSB)

*Hybrid method for computing periodic Schrodinger equation with band crossings*

We propose a hybrid method which combine the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with a periodic potential in the band-crossing case. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band crossing a BDTS method is used to capture the important inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver.