Math 2.0

Fluid mechanics Shock waves Problem

Shock tube problem

Builds on Riemann problem for gas dynamicsEuler equationsShock wave in gasesRarefaction wave

Problem

Consider the one-dimensional Euler equations of a polytropic gas with γ=1.4\gamma = 1.4 on the interval 0≤x≤10 \le x \le 1. At t=0t = 0 the gas is at rest, with density ρ=1\rho = 1 and pressure p=1p = 1 for x<0.5x < 0.5, and with ρ=0.125\rho = 0.125 and p=0.1p = 0.1 for x>0.5x > 0.5. Compute the solution at a later time, before any wave reaches the ends of the interval, and compare a numerical scheme with the exact solution.

Remarks

This is the Sod shock tube, a standard test for numerical methods for compressible flow, proposed by Gary Sod in a 1978 survey of finite difference schemes. It is a special case of the Riemann problem for gas dynamics, so its exact solution is known. The solution contains a rarefaction wave moving left, a contact discontinuity and a shock wave moving right. In the middle region the pressure is about 0.303130.30313 and the velocity is about 0.927450.92745. A good scheme must capture the shock sharply, keep the contact discontinuity from smearing too much, and produce no spurious oscillations; the comparison is often made at t=0.2t = 0.2.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Conservation law, concept, no evidence levelConservation lawconceptEntropy solution, concept, no evidence levelEntropy solutionconceptEuler equations, concept, no evidence levelEuler equationsconceptRarefaction wave, concept, no evidence levelRarefaction waveconceptShock wave in gases, concept, no evidence levelShock wave in gasesconceptRiemann problem, problem, no evidence levelRiemann problemproblemRiemann problem for gas dynamics, problem, no evidence levelRiemann problem for g…problemShock tube problem, problem, no evidence levelShock tube problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Conservation law concept
  2. Entropy solution concept
  3. Euler equations concept
  4. Rarefaction wave concept
  5. Shock wave in gases concept
  6. Riemann problem problem
  7. Riemann problem for gas dynamics problem
  8. Shock tube problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Sod_shock_tube
About this page
Tag9F314R2K5Q
URIm2://9F314R2K5Q
Kindproblem
WritingW0 stub
MSC76L05, 76N15, 65M08
StatusThis copy of the site has no build data. Run m2 status 9F314R2K5Q.
AboutRiemann problem for gas dynamics
UsesRiemann problem for gas dynamics, Euler equations, Shock wave in gases, Rarefaction wave
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.