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Godunov MHD and the Brio–Wu Shock Tube

Source: PHY653B Ch. 5

Intuition

Give up on kinetics and the difficulty does not disappear — it moves. MHD is only eight equations, but they are hyperbolic and nonlinear, so smooth initial data spontaneously produces discontinuities. The numerical problem becomes: how do you advance a solution across a shock without either smearing it into mush or generating oscillations that destroy the run?

Seven waves

Gas dynamics has three wave families (two sound waves and an entropy mode). Ideal MHD has seven: two fast magnetosonic, two slow magnetosonic, two Alfvén, and one entropy mode. The extra structure is what makes MHD Riemann solvers hard — and it is why MHD shock tubes are such demanding tests, since a single initial discontinuity can decay into a rich train of distinguishable structures.

Worse, MHD is non-strictly-hyperbolic: wave speeds can coincide, producing degeneracies that break Riemann solvers designed for gas dynamics.

Godunov's method

The idea: treat each cell as piecewise-constant, and at each interface solve the Riemann problem — the exact evolution of a single discontinuity — to get the flux.

\[U_i^{n+1} = U_i^n - \frac{\Delta t}{\Delta x}\left(F_{i+1/2} - F_{i-1/2}\right)\]

Because the flux comes from genuine wave propagation, shocks are captured sharply and in the right place, without artificial viscosity. The exact MHD Riemann solver is far too expensive, so practice uses approximate ones:

Solver Waves resolved Character
Rusanov / LLF 1 Bulletproof, very diffusive
HLL 2 (fastest each way) Robust; smears contact and Alfvén waves
HLLD 5 The modern default for MHD; resolves the full structure well
Roe 7 Sharpest; can fail at strong rarefactions without an entropy fix

The universal trade: more resolved waves means sharper features and more ways to fail. Production codes routinely fall back to HLL locally when HLLD misbehaves.

Second-order accuracy needs reconstruction (MUSCL, PPM) with a slope limiter, which is where Godunov schemes earn their reputation: the limiter suppresses oscillations near discontinuities at the cost of reducing to first order exactly there.

Brio–Wu: the benchmark

The standard MHD shock tube. A single initial discontinuity decays into a structure containing a fast rarefaction, a compound wave (a shock attached to a rarefaction — a peculiarly MHD object with no gas-dynamic analogue), a contact discontinuity, a slow shock and another fast rarefaction.

It is the benchmark because it is unforgiving: get the Riemann solver, the limiter, or the divergence handling wrong and the wave positions or amplitudes shift visibly. Every MHD code publishes its Brio–Wu.

Measure it, do not eyeball it. Compare wave positions against the reference, and run a convergence study — with a caveat: a solution containing discontinuities converges at first order in \(L_1\) no matter how high-order your scheme is, so verify formal order on a smooth problem and use Brio–Wu to check structure.

Orszag–Tang

The 2-D companion: smooth, benign initial data that develops an intricate network of interacting shocks. Where Brio–Wu tests the Riemann solver in isolation, Orszag–Tang tests whether the whole scheme survives shock–shock interaction, and it is the standard visual signature of a working MHD code.

Common mistakes

  • Reporting high-order convergence on a shock problem. \(L_1\) convergence is first order at a discontinuity regardless of the scheme.
  • Ignoring the divergence constraint. \(\nabla\cdot\mathbf{B}\) errors produce spurious forces parallel to \(\mathbf{B}\) and can wreck a Brio–Wu profile.
  • Blaming the Riemann solver for limiter problems. Oscillations near discontinuities are usually the reconstruction, not the flux.

Knowledge graph position

Prerequisites: MHD, hyperbolic conservation laws, shock waves. Leads to: divergence constraint, resistive MHD, reconnection simulation, the kink instability in flux ropes.

Quiz

Q1 (conceptual). Why does ideal MHD have seven wave families where gas dynamics has three?

Answer

The magnetic field adds degrees of freedom: two Alfvén waves (transverse, driven by magnetic tension) and a splitting of the sound wave into fast and slow magnetosonic branches, plus the entropy mode — seven in total. The extra structure, and the degeneracies where speeds coincide, are what make MHD Riemann solvers hard.

Q2 (conceptual). Your second-order MHD scheme shows first-order \(L_1\) convergence on Brio–Wu. Bug or expected?

Answer

Expected. A discontinuous solution limits \(L_1\) convergence to first order for any scheme, because the error is dominated by the finite number of cells spanning the jump. Verify formal order on a smooth problem (a circularly polarised Alfvén wave, or MMS) and use Brio–Wu for structure instead.

Q3 (MCQ). HLLD is preferred over HLL for MHD because it:

  • (a) is faster
  • (b) resolves five waves rather than two, so contacts and Alfvén waves are not smeared
  • (c) never fails
  • (d) does not need a divergence-cleaning scheme
Answer

(b). HLL keeps only the two fastest waves and diffuses everything between them. HLLD restores the rotational and contact structure — at the cost of more failure modes, which is why codes fall back to HLL locally.