Shock Capturing
Shock-capturing schemes compute discontinuous solutions of conservation laws automatically, without tracking discontinuity locations explicitly.
Letting the scheme find the shock
Nonlinear hyperbolic conservation laws, such as those of compressible flow and MHD, develop discontinuities (shocks and contact surfaces) even from smooth initial data. Shock-capturing methods handle these without any explicit knowledge of where the discontinuities are: the discontinuity emerges naturally as a steep numerical transition spread over a few cells, and its speed and jump are correct provided the scheme is conservative.
This contrasts with shock-tracking (front-tracking) methods, which represent discontinuities as explicit moving interfaces. Capturing is far more robust for complex flows with many interacting or newly forming shocks, at the cost of a few cells of smearing at each front.
Why conservation is essential
The Rankine-Hugoniot conditions relate the jump across a shock to its speed. A discrete scheme reproduces the correct shock speed only if it is written in conservative (flux-difference) form; the Lax-Wendroff theorem guarantees that a convergent conservative scheme converges to a genuine weak solution with correct jumps. Non-conservative schemes can converge to wrong shock speeds, a serious and hard-to-detect error.
Controlling oscillations and entropy
High-order schemes need oscillation control near shocks, supplied by flux limiters, TVD constraints, WENO weighting, or artificial viscosity. An additional entropy condition selects the physically admissible weak solution among mathematically valid ones, preventing unphysical expansion shocks; entropy-fix and entropy-stable schemes enforce it.
- Discontinuities emerge as steep numerical transitions
- Conservative form required for correct shock speeds
- Oscillation control via limiters, TVD, WENO, or viscosity
- Entropy condition selects the physical weak solution
Shock-capturing techniques are essential in fusion-relevant MHD and compressible modeling, where fast magnetosonic shocks and steep fronts arise and must propagate at the correct speeds.