Van Leer & HLLC Part 2

General Relativity (GR)

Moving from special relativity to general relativity this week. The homework was pretty long so I spent two days on that.

HLLC Riemann Solvers

I’m about half done writing an HLLC Riemann solver, specifically the one from Batten et al. 1997. The details of the solver can be found in my previous post HLLC Algorithm. I’ve had to make some tweaks to the implementation of the exact Riemann solver, namely moving the position argument to the end of the parameter list and making it optional. In numerical simulations it’s always zero and doesn’t usually show up in approximate solvers so now I can just neglet it when I don’t need it.

Van Leer Integration

With the help of my advisor I found the issues with my Van Leer integrator. There were two separate issues:

1. Since the Van Leer integrator requires a first order and then a second order reconstruction you need a third ghost cell. I was just updating the real part of the grid.
2. The first order reconstruction and Riemann solve serve the purpose of updating the grid half a time step and the plain second order reconstruction does the second half of the time step. Instead of using a plain second order reconstruction I was using the same reconstruction algorithm from my CTU integrator which contains characteristic tracing. That characteristic tracing is what does the half time step update in the CTU integrator. Since I was doing it after already doing a half time step update I was effectively updating past one time step which is bad.

Fixing these two issues and changing the resolution and CFL number to more realistic values (1000 cells and 0.4) fixed all the issues I was having as you can see in the plot below.

The Sod Shock Tube using a Van Leer algorithm and PLM reconstruction