Van Leer and HLLC

General Relativity (GR)

Still working on special relativity this week. Spent most of Monday and Tuesday morning on homework. We did derive the relativistic rocket equation and that was fun.

HLL Family Riemann Solvers

I read the chapter of Toro on HLL family Riemann solvers since we’re planning to implement a HLLD Riemann Solver for MHD. The solver is way simplier than Toro’s exact solver (thankfully) and I think I’ll implement it next week. The general algorithm for the HLLC Riemann Solver is:

  1. Estimate the pressure in the Star region. A PVRS estimate is usually good enough
  2. Estimate the wave speeds
  3. Based on the wave speeds figure out which of 4 states we’re in
  4. Compute the HLLC flux (there are two variants, one that assures constant pressure accross the contact wave and one that doesn’t)
  5. Profit :)

Van Leer Integration

I’ve finished converting my hydro code in hydro-sandbox to use a Van Leer integrator instead of a CTU integrator so that I can use the VL+CT method for MHD detailed in Stone & Gardiner 2009.

Currently it is running and giving broadly correct results but is way too diffusive and I’m not sure why. You can see in the first video how clear the shock is using the CTU algorithm. In the second video it’s significantly more diffuse and I’m not sure why. Both are using piecewise linear methods to compute the interface states.

The Sod Shock Tube using a CTU algorithm and PLM reconstruction

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

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