Riemann Solvers
Riemann solvers compute the flux between two states across a discontinuity by resolving the resulting wave structure, exactly or approximately.
The building block of upwind schemes
A Riemann problem is a conservation law with initial data consisting of two constant states meeting at a single discontinuity. Its solution is a self-similar fan of waves: shocks, rarefactions, and contact discontinuities moving at characteristic speeds. Riemann solvers compute this wave structure, or enough of it to evaluate the flux at the interface, which is the core operation of every Godunov-type finite-volume scheme.
By resolving which waves travel in which direction, the Riemann solver supplies exactly the upwind information needed for a stable, conservative flux, encoding the physics of the equations into the numerics.
Exact and approximate families
The exact solver iterates for the intermediate pressure and velocity in the star region; it is accurate but costly and specific to the equation set. Approximate solvers trade some accuracy for speed and generality. The Roe solver linearizes the flux using a specially averaged Jacobian; HLL assumes a two-wave structure with estimated fastest speeds; HLLC restores the contact wave; HLLD adds the rotational and slow waves needed for MHD.
Robustness and fixes
Approximate solvers can fail in edge cases: the Roe solver may admit unphysical expansion shocks without an entropy fix, and low-density or strong-rarefaction states can produce negative pressures. Positivity-preserving variants and entropy fixes guard against these failures, which is why robust solvers like HLLC and HLL are popular defaults despite slightly more diffusion.
- Riemann solution is a self-similar fan of waves
- Roe: linearized flux via a special average Jacobian
- HLL/HLLC/HLLD: wave-speed-estimate solvers of rising fidelity
- Entropy fixes and positivity preservation for robustness
In MHD simulation the HLLD solver, which resolves the full seven-wave structure, is a common choice for accurately capturing the fast, slow, and rotational discontinuities relevant to magnetized plasma dynamics.