T3: The Riemann problem:
try it out on your own

Before the tutorials:

As discussed in the Lecture, the Riemann Problem is the key pillar of Eulerian hydrodynamics and can not be solved fully analytically. In the lecture, we solved a special case of it. So, start with this picture taken from Simulation Techniques for Cosmological Simulations. Now, think about:

• How can we use the particle distributions from last time to initial conditions for the problem above?
• How are different densities represented?
• In the Rieman problem we used the pressure P, in the initial condition we need internal energy per unit ass u. How can i get u for a given P?
• How can I verify the solution I get from the simulation?

Also, remember that we had the equation with and that determines the pressure at the contact discontinuity (the pressure in sectors 3 and 4). Think about:

• How can you simply solve the this equation?
• How can you check it against the simulation performed?

During the tutorials:

Run a small hydro test problem for a jump in pressure and density. With time, a situation similar to shown above should emerge, where you should be able to clearly identify the 5 regions marked above.

Make the movie of the simulation, similar to last time, and then, compute the solution of the Riemann problem and overplot them over the time evolution. You can start with the compute pressure P5, and then try to recompute some other values in the different regions as well.

Simple, 2D example

As in the previous tutorial (This time we start with a 2D setup), get a copy of the code and the configuration file Config.sh. But before compiling, edit the Config.sh file so that it contains:

• PERIODIC
• NOGRAVITY
• WENDLAND_C2_KERNEL
• TWODIMS
• LONG_X=40
• LONG_Y=1
• LONG_Z=1

Do not forget the to compile the make -j command before you start the simulation.

As in the first tutorial, copy the parameter file box.param and modify it to a new and reasonable unit system and adapt the times for the outputs according to resonable values:

• PeriodicBoundariesOn 1
• TimeMax 5.0
• TimeBetSnapshot 0.1
• UnitLength_in_cm 1.0
• UnitMass_in_g 1.0
• UnitVelocity_in_cm_per_s 1.0
• SofteningGas 0.2
• SofteningGasMaxPhys 0.2
• Remember: Here we do not use Gravity, so we can use a relatively large softening for the particles, as the timesteps should be defined by hydro-dynamic properties!

First Step

Use the prepared ICs to do your simulation:

• wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T03/box.ic

Now run the simulation and look at the result. Note that you can already watch the results from the existing snapshots while the simulation is still running. You can attach a new command prompt to the running container by opening a new shel and execute docker exec -it opengadget3_container bash Try to modify your plotting scripts from the first tutorial T01 to do a simple plot showing the different values along the x-axis.

Second Step

As you might realize, the numerical solution looks quite noisy, so you can start to apply what you learned in T03.

• First, try to improve the results by using close packing in 2D instead of the simple grid you used so far.
  Hint: Think what distribution shifting the particle rows by 0.25 to left and right alternately gives?
• Second, try to see how results improve by different kernels.
  Hint: You can first try to use the WENDLAND_C4_KERNEL and then the WENDLAND_C6_KERNEL.

Now, as the results look much better, there is blip in the pressure left. Check if it is already present in snap_000 and if so, think about where it could come from. You can try to run the simulation switching additional ARTIFICIAL_CONDUCTIVITY on.

Third Step

Now you can play around with different initial values for left and right values for density and pressure.

Programming goals for T4:

Goal of this tutorial is that you

• learn better how to create non-uniform initial conditions.
• learn how to compute other hydrodynamic quantities (like the pressure P).
• learn to produce simple plots of other quantities.

Solutions

• To solve the Riemann problem you can create a table of pressures and then plot the left hand side (lhs) and the right hand side (rhs) of the above equation in a single plot. Just use the values from the initial conditions you created. Then you can find the value of the pressure P where the two curves cross each other:
• Example for script to set up simulations:
  • wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/Python/ic_header
    (if not already done before)
  • wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T03/setup_grid.py
  • python3 setup_grid.py
• Example(s) for visualizing the results:
  • wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T03/show_simple.py
  • python3 show_simple.py

Additional exercise for experts/very interested students (voluntary)

As you use a lagrangian scheme, you can follow the fluid elements explicitly. All particles have an unique identifyer ID, so you can use that to color code the particles.

Can you see some interesting details in this plot? Try to see if you can get a intuitve understanding what the interplay of the motion of the featurs and the change of the color of the features mean for the flow of the gas!

Useful commands

• More informations on the OpenGadget3 containers
• Start your docker session:docker run -it -v :/mnt --name opengadget3_container opengadget3:XXX
• Connect from a second shell to a running docker session:docker exec -it opengadget3_container bash
• Get the simulation code:cp -r ~/OpenGadget3 .
• Copy the config file: wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/Config.sh
• Copy the parameter file: wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/box.param
• Setup the environment for compiling the code: source Build/Container_build.sh
• Compile the code: make -j
• Run a simulation: mpiexec -np 2 OpenGadget3/OpenGadget3 box.param