LQG-BH-Collapse

This code implements the original formulation of the Weighted Essentially Non-Oscillatory [1] (WENO) to solve the Loop Quantum Gravity equations of a collapsing star under the assumption of spherical symmetry. It solves the equation

$$ \begin{cases} &\partial_t B(x, t) = - \partial_x \left[ \frac{1}{2} x^3 \sin^2 \left( \frac{B}{x^2} \right) \right] + \frac{1}{2}\varepsilon^b \\ &\partial_t \varepsilon^b (x, t) = - \partial_x(\varepsilon^b)\sin\left( 2\frac{B}{x^2} \right) \end{cases} $$

for various initial condition. The simplest possibility is Oppenheimer-Snyder collapse:

$$ \begin{cases} &B(x, 0) = - \frac{x^2}{2} \arccos\left( 1 - 4\frac{M(x)}{x^3} - 2 \frac{\varepsilon^b}{x^2} \right), \qquad M(x) = 4\pi\int_0^x \rho(y)y^2 dy \\ &\varepsilon^b(x,0) = - \frac{x^2}{a_0^2} \theta(x - r_0) - \frac{r_0^2}{a_0^2} \left(1 - \theta(x - r_0)\right) \end{cases} $$

where

$$ \rho(x) = \frac{m}{\frac{4}{3}\pi r_0^3} \left( 1 - \theta(x - r_0) \right), \quad \theta = \begin{cases} 0 & \text{ if } x \lt r_0\\ 0.5 & \text{ if } x = r_0\\ 1 & \text{ if } x \gt r_0 \end{cases} $$

but arbitrary density profiles are possible.

A Python module to manage the simulations is available here.

How to install

To install the Fortran code run the bash script make.sh using the following command

./make.sh

It creates a folder named build that contains a copy of the source code as present at the moment of compilation, the Makefile and the executable named run. It will also create a subfolder named output that will be the default target for the output routines and copy the standard parameter file that can be customized. It requires the HDF5 [2] package to be installed and on the PATH environmental variable.

To not use HDF5 compile using the Makefile with the NOHDF5=1 option

make NOHDF5=1

or modify by hand the make.sh file.

The name and location of the build folder can be customized providing the make.sh script a path as an argument

./make.sh path/to/custom/build/folder

To remove an existing build folder and all its content the dedicated command is

./make.sh remove path/to/build/folder

To run the code there are two optional arguments to be given:

./run [path/to/parameter/file.dat] [path/to/output/folder]

If they are not given the standard paths will be used; they are . and ./output respectively. Please note the code assumes the executable is run from inside the build folder; if executing it from elsewhere these two arguments become mandatory.

Test Suite

The test suite includes a set of tests to verify the functionality and correctness of the project. To run the tests, follow the steps below:

1. Open a terminal and navigate to the project's root directory.

2. Change the current directory to the tests folder and compile using the command:

make tests

3. Run the test suite using the following command:

./tests

4. After running the tests, it is recommended to clean up the generated files by running the following command:

make clean

Description of the parameter file

The parameter file contains the following information:

• ID: Index representing the initial condition

  • Explanation: Identifies the specific initial condition to be used in the simulation.
    • ID = 0, full dynamics for Oppenheimer-Snyder collapse, closed universe
    • ID = 1, post bounce dynamics for Oppenheimer-Snyder collapse
    • ID = 2, post bounce dynamics with peaked initial density function
    • ID = 3, full dynamics starting from atan density profile nd smooth $e^b$
    • ID = 4, full dynamics for Oppenheimer-Snyder collapse, flat universe
    • ID = 5, full dynamics for Oppenheimer-Snyder collapse, open universe
  • Range of values: Any integer value between 0 and 5, inclusive.

• Total simulation time: Total duration of the simulation

  • Explanation: Specifies the length of time the simulation will run.
  • Range of values: Any positive floating-point number.

• Characteristic radius, r0: Characteristic radius used in the simulation

  • Explanation: Determines the characteristic size of the system.
  • Range of values: Any positive floating-point number.

• Scale factor, a0: Scale factor used in the simulation

  • Explanation: Determines the overall curvature of the system.
  • Range of values: Any positive floating-point number.

• Mass: Mass value used in the simulation

  • Explanation: Specifies the mass of the system.
  • Range of values: Any positive floating-point number.

• Order of WENO: Order of the Weighted Essentially Non-Oscillatory (WENO) scheme

  • Explanation: Determines the accuracy and stability of the numerical method.
  • Range of values: 2 (order 3 is not implemented -yet-).

• Furthest grid point: Furthest point on the computational grid

  • Explanation: Specifies the farthest extent of the computational domain.
  • Range of values: Any positive floating-point number.

• Grid spacing: Spacing between grid points

  • Explanation: Specifies the distance between neighboring grid points.
  • Range of values: Any positive floating-point number.

• Save output every ## iterations: Frequency of saving output

  • Explanation: Determines how often the simulation results are saved.
  • Range of values: Any non-negative integer value. Use 0 to disable saving.

• Print output every ## iterations: Frequency of printing output

  • Explanation: Specifies how often the simulation results are printed.
  • Range of values: Any non-negative integer value. Use 0 to disable printing.

• Number of threads: Number of threads for parallel processing

  • Explanation: Determines the degree of parallelism in the simulation.
  • Range of values: Any positive integer value.

Sources

[1] Xu-Dong Liu, Stanley Osher, Tony Chan, Weighted Essentially Non-oscillatory Schemes, Journal of Computational Physics, Volume 115, Issue 1, 1994, Pages 200-212, ISSN 0021-9991, https://doi.org/10.1006/jcph.1994.1187.

[2] https://www.hdfgroup.org/solutions/hdf5/