Exact MHD Riemann Solver
Exact solutions of Riemann problems of one-dimensional ideal magnetohydrodynamics (γ-law gas), following M. Torrilhon, Exact Solver and Uniqueness Conditions for Riemann Problems of Ideal Magnetohydrodynamics, SAM Research Report 2002-06, ETH Zürich.
Supported initial states
Checked before any solve. Field thresholds refer to |B|/√p with p the pressure of the same state.
| quantity | condition | otherwise |
|---|---|---|
| all values, γ | finite, γ > 1 | invalid input |
| density, pressure | ρ > 0 and p > 0 on both sides | invalid input |
| normal field Bx | equal on both sides to within 10⁻¹²·max(|Bx|, 1) | invalid input |
| jump ratios | 10⁻⁶ ≤ ρR/ρL ≤ 10⁶ and 10⁻⁶ ≤ pR/pL ≤ 10⁶ | refused |
| velocities | no restriction | – |
| case | normal field | transverse field | solver |
|---|---|---|---|
| vanishing normal field | |Bx|/√p ≤ 10⁻¹⁰ | any, including Bt = 0 | quasi-Euler (Bx = 0) |
| regular | |Bx|/√p > 10⁻¹⁰ | |Bt|/√p ≥ 10⁻⁴ on the larger side and ≥ 10⁻⁶ on the smaller side | regular |
| otherwise | |Bx|/√p > 10⁻¹⁰ | either condition violated | refused (switch-on/off) |
The quasi-Euler threshold is bn_euler in the advanced options. A problem that passes can still end as RegularLimit when its solution needs a limit of the regular waves: vacuum, a middle state with |Bt|/√p < 10⁻⁸, or an intermediate or compound wave. With both transverse fields close to their thresholds a few percent of problems end as NoConvergence.
| ρ | vx | vy | vz | By | Bz | ∠Bt | p |
|---|
About
This page computes the exact solution of the Riemann problem of one-dimensional ideal magnetohydrodynamics (MHD) for a γ-law gas: two constant states separated at x = 0, which evolve into fast and slow shocks or rarefaction fans, rotational (Alfvén) discontinuities and a contact. Typical use is the verification of numerical MHD schemes against shock-tube problems such as Brio–Wu. Every solution passes independent checks (Rankine–Hugoniot, Lax and entropy conditions) before it is shown; the computation runs entirely in your browser.
Method: M. Torrilhon, Exact Solver and Uniqueness Conditions for Riemann Problems of Ideal Magnetohydrodynamics, SAM Research Report 2002-06, ETH Zürich, 2002; see also M. Torrilhon, Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics, J. Plasma Phys. 69(3), 253–276 (2003), doi:10.1017/S0022377803002186. Source code: ExactMHDRiemannApp (this page, TypeScript) and the Julia package ExactMHDRiemannSolver.jl.