Works and Days

Presidents Summer Fellowship, Modeling Fluid Dynamics Part 3, Qiaoyu Yang

President’s Summer Fellow Qiaoyu Yang ’16, mathematics major, is testing a probabilistic particle model for studying fluid dynamics with Prof. Aleksandar Donev at the Courant Institute in New York City.

This summer, I worked with Prof. Aleksandar Donev in Courant Institute to perform particle simulation for chemically reactive fluid. In the following I will try to explain the essentials of the project.

Our research problem is to model reactive fluid. Traditionally, fluids’ dynamics are modeled mostly by differential equations. However, in our case, because of the chemical reactions involved, some assumptions about the fluid is very different from reality and this makes the results described by differential equation to be inaccurate. Therefore, we need to use some other methods.

In such a case, usually mathematicians/scientists would use microscopic modeling methods, which include molecular dynamics (MD), direct simulation Monte Carlo (DSMC), and so on. However, they don’t really work well in our problem because of high computational cost or limited applicability. Therefore, we chose to turn to the reaction diffusion Master equation (RDME) model. RDME model describes the behavior of particles undergoing both reaction and diffusion processes. Diffusion process is relatively easier to implement and its time complexity is much lower than that of reaction process. Hence, our main focus is on the reaction part. Instead of processing reactions in deterministic way, which is what MD does, we process stochastic reactions. By doing so, we won’t get results accurate at the particle scale but the global behavior is very accurate.

The way we implement RDME is to first create a box, where the reaction domain is in. Then, we divide the box into regular cells with pre-calculated size and only allow reactions to happen for particles in the same cell. However, in doing so, we introduced artifacts. Specifically, because we only allow particles in the same cell to react, we would miss reactions between particles in neighboring cells and therefore create a difference between the boundary and the center of each cell. This is not physical since in reality there does not exist “cells” and therefore the difference between distribution of particles near boundary and that of particles near center of the cell is a wrong result.

To fix this problem, we use inspiration from isotropic DSMC, some algorithm developed by Prof. Aleksandar Donev several years ago. Instead of allowing only same-cell reactions, we now allow reactions for particles in neighboring cells. By choosing appropriate cell size, we can take into consideration all the reactions missed in RDME. However, by doing so, we can no longer process reactions cell-wise, which would cause many difficulties in the scheduling of reactions. We solve the problem by using a linked list to store the particles and using priority queue to record the events. We did some numerical tests to show that the code is doing the right thing.

We are almost done with IRDME, although we may consider other options of implementation in the future so as to improve efficiency of the code. So far, we have done numerical tests on RDME and the result, when compared with earlier researches, indicated that the code is generating accurate data. We have also done some tests on IRDME to check that it behaves in correct ways in certain aspects. After we finish all the code, we will run complete tests on IRDME and compare it with RDME to see the difference.

After the summer, I am still working remotely with Aleks, that is, my supervisor at Courant Institute. It’s really amazing how helpful Aleks is. The experience is truly fulfilling and certainly I feel like I get what I wished to attain from this research. Again, I am very grateful for Mr. Dan Greenberg, who kindly provided the funding for President’s Summer Fellowship, with which I got to work in Aleks’ group. I am currently writing my thesis based on this project and hopefully we will achieve more in the future. 

Tags: psf, presidents summer fellowship, mathematics, math, programming, computation, fluid dynamics, computer science