Numerical Analysis of the Jeans Instability

Size: px

Start display at page:

Download "Numerical Analysis of the Jeans Instability"

Nickolas Rodney Peters
7 years ago
Views:

1 June 15, 2010

2 Background Goal Refine our understanding of Jeans Length and its relation to astrophysical simulations. Currently, it is widely accepted that one needs four cells per Jeans Length to get adequate resolution in a simulation, but this itself was determined only for very specific cases. Any refinements to this criterion could make a significant impact on astrophysical simulations.

Currently, it is widely accepted that one needs four cells per Jeans Length to get adequate

3 Background Project Members Prof. Robert Fisher: Kelly Truelove: Avinash Kumar: Advisor Consultant Graduate Student Research Assistant Mr. Truelove was the lead author of the paper Self-Gravitational Hydrodynamics With Three-Dimensional Adaptive Mesh Refinement: Methodology and Applications to Molecular Cloud Collapse and Fragmentation, which was printed in the the March 1998 issue of the Astrophysics Journal. His paper is the inspiration for our current work. The equations from the next slide are presented in Appendix B of his paper.

Truelove was the lead author of the paper Self-Gravitational Hydrodynamics With Three-Dimensional Adaptive Mesh Refinement:

4 Gravitohydrodynamics Simulations Isothermal Euler Gravitohydrodynamic Equations The isothermal Euler equations of gravitohydrodynamics are being used in this research Continuity Equation δρ δt + δ(ρv) δx = 0 Conservation of Momentum ρ δv δv + ρv δt δx = δρ c2 s δx ρδφ δx Gravitational Potential (Poisson Equation) δ 2 ρ δx 2 = 4πGρ For the simulation, these equations are discretized and solved for points on a grid.

0 Conservation of Momentum ρ δv δv + ρv δt δx = δρ c2 s δx ρδφ δx Gravitational Potential (Poisson

5 Gravitohydrodynamics Simulations The Original Discretizations Continuity Equation ρ n+1 j Conservation of Momentum v n+1 j = ρ n j t x ((ρv)n j (ρv) n j 1) = v n j t x (((v)n j (v) n j 1) c2 s ρ ((ρ)n j (ρ) n j 1) ((φ) n j (φ) n j 1))

of Momentum v n+1 j = ρ n j t x ((ρv)n j (ρv) n j 1) = v n

6 Debugging Problems with the Code Prior to my arrival on the project, Avinash had written the simulation. It only worked for supersonic speeds in the positive direction. Prof. Fisher recommended module testing. Avinash had tested the Poisson solver prior to my arrival. I was asked to module test the other two sections.

It only worked for supersonic speeds in the positive direction. Prof.

7 Debugging Module Testing To help isolate the effects of any bug, we assumed an isothermal situation, constant pressure, and no gravity, and a periodic boundary between two different regions.

8 Debugging Continuity Equation In this module test, we were trying to see if the modeling of density was working correctly. To isolate velocity s effect, it was set constant throughout the entire grid. I was asked to simulate the first timstep. At first, I did not locate an obvious source of the instability.

To isolate velocity s effect, it was set constant throughout the entire grid.

9 Debugging Conservation of Momentum For this simulation, I was asked to set up intial conditions that modeled a shock. I was asked to use a Mach number of my choice (preferably close to 1) and then set the following. c iso = 1.0 u 1 = c iso M p 1 = 1.0 p 2 = p 1 M 2 u 2 = u 1 M 2 During this test, obvious instabilties were present, but I could not figure out why.

I was asked to use a Mach number of my choice (preferably close to 1) and then set the

10 Debugging Correcting the Instability During a discussion of my results during a teleconference with Mr. Truelove, he suggested that Mach scaling could be at fault. While I was looking into this possibiliy, I mentioned my results to Prof. Sigal Gottlieb. She suggested that the true problem with the code was the improper use of an upwind method (which our finite differencing was). After a meeting of Prof. Fisher, Prof. Gottllieb, and myself, it had been firmly established that our method was unconditionally unstable for any conditions other than supersonic speeds in the positive direction. A new method that could handle information from both directions was necessary.

She suggested that the true problem with the code was the improper use of an upwind method (which our finite differencing was). After a meeting of Prof. Fisher, Prof.

11 Moving Forward Prof. Fisher has recommended that we proceed by replacing our finite differencing method with a first-order Godunov method. This is considerably more complicated, but is supposed to be stable in the situations that we are studying. I have been asked to write the new code and combine it with Avinash s working Poisson solver. As a first step, I have written a code to solve the Riemann Problem, a requirement for using the Godunov method. To do so, I needed to spend several days reading about linear algebra, particularly eigenvalues and eigenvectors, so that I would understand the code that I was writing.

I have been asked to write the new code and combine it with Avinash s working Poisson solver.

12 Moving Forward Today, I have produced a program that solves the Riemann problem across a grid. I have also attempted to include a period boundary condition.

Advanced CFD Methods 1

Advanced CFD Methods 1 Prof. Patrick Jenny, FS 2014 Date: 15.08.14, Time: 13:00, Student: Federico Danieli Summary The exam took place in Prof. Jenny s office, with his assistant taking notes on the answers.