Perimeter Scholars International
Explorations in Gravitational Physics
Numerical Relativity

MW Choptuik, L Lehner & F Pretorius

Quick Links: [ Project 1 : Project 2 : Project 3]

Lecture Notes

Week 1: Solution of Classical Field Equations Using Finite Difference Techniques (MW Choptuik)

1. Solving the wave equation using finite difference techniques [PDF]
2. The 3+1 approach to the Einstein equations [PDF]
3. Time dependent spherically symmetric (SS) spacetimes and the SS Einstein-massless-Klein-Gordon system [PDF]
   
4. Project 1 handout [PDF]
   

Week 2: Much Ado About Hydrodynamics and Related Problems (L Lehner)

1. Overview and Burger's equation (includes Project 2) [PDF]
2. General Relativistic Hydrodynamics and beyond [PDF]
   

Week 3: Miscellaneous topics (F Pretorius)

1. Lecture 1: Overview [PDF]
2. Lecture 4: AMR [PDF]
   

Background/Reference Material and Related Software  

Project 1: Scalar Wave Evolution on a Schwarzschild Background

Project 2: Ultrarelativistic Hydrodynamics

• Project handout (Secs. 5,6 and 7 of notes: begins at page 10) [PDF]
• Code for solution of Burger's equation. Distribution: burgers.tar.gz
  
1. burgers_rnpl
2. init_q.inc
3. step.inc
   
4. calc_A.f
5. calc_eta.f
6. calc_f.f
7. calc_FF.f
8. calc_flux.f
9. calc_lambda.f
10. calc_omega.f
11. indep_res.f
12. minmod.f
13. norm.f
14. recos_qL.f
15. recos_qR.f'
16. update_boundary.f
17. update_q.f
• Code for exact solution of Burger's equation. Distribution: burgers_exact.tar.gz
• Animations made with burgers_exact and burgers.  Three distinct initial data sets are used, two Riemann (shock tube)  configurations and one generic smooth initial data (Gaussian).  Specifically
• R1: Riemann problem 1: Shock:  q_L=1.0  q_R=0.1
• Exact solution [MPEG]
  
• Exact solution & Roe solution [MPEG]
  
• R2: Riemann problem 2: Fan: q_L=0.1  q_R=1.0
• Exact solution [MPEG]
  
• Exact solution & Roe solution [MPEG]
  
• G: Gaussian initial profile
• Roe solution applied to q_t + (1/2 q^2)_x = 0 [MPEG]
  
• Roe solution & first-order upwind applied to q_t + q q_x = 0 [MPEG]
  
• Roe solution & first-order upwind applied to q_t + (1/2 q^2)_x = 0 [MPEG]
  
• Roe solution & second-order upwind applied to q_t + (1/2 q^2)_x = 0 [MPEG]
  
• Code for exact solution of Riemann problem for ultrarelativistic fluid. Distribution: ultra_exact.tar.gz
• Animation made with ultra_exact and the current key for the problem, ultra as implemented by Pretorius/Olabarietta
• R1: Riemann problem 1: Shock/Fan q_L=1.0  q_R=0.1: Pressure [MPEG]
  
• HINTS for groups having difficulties with the implementation.
  1. Version of properly functioning code's step.inc with numerous calls to dvdump (see below) added to provide a detailed trace of top-level steps in the evolution algorithm. You should try to instrument your implementation of step.inc with identical calls to dvdump in order to easily compare your results with ours.
  2. Initial data file id0 for an evolution consisting of a single time step (apart from the initial time), on a mesh with 16 total grid cells, and for shock tube data with rho_R = 0.1, rho_L = 1.0, v_R = v_L = 0. Again, to compare with our results you will need to ensure that the parameter file that you use matches this one in all relevant respects. 
  3. Output file fort.80 containing debugging trace from dvdump, using above parameter file, and the functioning code.
     
  4. Source code for dvdump.f. Download this file into your code directory, and, in the Makefile, add dvdump.o to the list of object files on which the executable depends. Ask an instructor if you have any questions concerning this.

Project 3: Teukolsky Equation

• Project handout [PDF]
• Code for Azimuthally-Separated Wave Equation in Spherical Polar Coordinates. Distribution: wave2d_sph.tar.gz
  
1. wave2d_sph_rnpl
2. Makefile(sample)
3. id0(m=0)
4. id0(m=1)
5. id0(m=2)
• DV overview
  
• Generating MPEG animations using DV
  
• Instructions on use of DV and xvs for extraction of QNFs from simulation results