SPH 1D Sod Shock Tube

What you are seeing: a one-dimensional smoothed-particle hydrodynamics (SPH) simulation of the Sod shock tube. The membrane at $x = 0.5$ separates a high-pressure left state ( $\rho = 1$ , $P = 1$ , $v = 0$ ) from a low-pressure right state ( $\rho = 0.125$ , $P = 0.1$ , $v = 0$ ). At $t = 0$ the membrane is removed.

The exact Riemann solution has three waves: a left-moving rarefaction fan (where the gas accelerates and rarefies), a contact discontinuity in the middle (where density jumps but pressure and velocity are smooth), and a right-moving shock (where density, pressure, and velocity all jump). The SPH method represents the fluid as 360 Lagrangian particles smoothed by a cubic-spline kernel; the artificial viscosity is what allows it to capture the shock.

Figure 1. SPH Sod shock tube. Method: cubic-spline kernel SPH with Monaghan-Gingold artificial viscosity (alpha=1, beta=2).

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WHAT TO TRY

Let the membrane burst and watch the three waves separate: a rarefaction fan runs left into the high-pressure gas, a contact discontinuity drifts right, and a shock runs furthest right. The density, velocity and pressure panels show all three.
Read the conserved quantities: the SPH particles carry mass exactly and the total energy drifts by only a percent, the honest accuracy check for the scheme on this classic Riemann test.
Switch the coloured field: the same particles re-coloured by density, velocity or pressure reveal which wave each feature belongs to, the contact shows in density but not pressure.