A dam break

The shallow-water solver had no reference behind it at all. This case gives it one, and like the shock tube the reference is arithmetic rather than somebody else’s experiment: an ideal dam break over a flat, frictionless bed has a closed-form answer, so there is nothing to transcribe from a paper and nothing that can be mis-transcribed.

It also has a number a reader can check without running anything. Whatever the reservoir depth, the water directly over the dam site settles instantly at exactly four ninths of it, and stays there.

Water at rest, depth h₀ to the left of a dam at x = 0, the dam removed at t = 0. With c₀ = √(g h₀) the state depends on x and t only through the similarity variable s = x/t:

    s <= -c0        h = h0                      u = 0
    -c0 < s < 2c0   h = (2c0 - s)^2 / (9 g)     u = (2/3)(s + c0)
    s >= 2c0        h = 0                       u = 0

The harness derives it rather than quoting it: the fan is the simple wave on which the Riemann invariant u + 2c is constant at its upstream value 2c₀, and on which the other characteristic speed u − c equals s. Solving those two together gives the formulae above, and a self-check confirms the result satisfies the shallow-water equations in similarity form to machine precision.

Putting s = 0 into the middle line gives the two numbers this page is measured on:

This is the one place where the reference and the product are solving different problems, and it is stated rather than tuned away.

Ritter’s other two assumptions do hold here exactly, and the harness asserts both rather than assuming them. The solver has no friction term at all — no Manning, no Chézy, no Laplacian in its momentum equation — so there is nothing to switch off. And the bed elevation is an optional field the product does not write, so the bed is flat; the harness refuses to continue if one ever appears, and keeps a separate fixture showing what a sloping bed looks like. Coriolis is off in the staged gravity properties, and that is asserted too.

One agreeing depth is one number that happens to be close. What makes this verification is that the error over the fan falls when the mesh is refined.

The order sits below one, and that is the expected behaviour rather than a shortcoming: this solution has a shock in it, and a shock reduces any scheme to first order locally. The fan norm still falls monotonically on both fields and on every pair of meshes.

Three things, and none of them is hidden in the numbers above.

• Ritter’s dry bed has not been run, and cannot be. The solver cannot construct a velocity where the depth is zero. Every number here is from a case with a 1 mm film downstream, and the whole-domain solution for that case is Stoker’s, not Ritter’s. Ritter is exact only over the fan, and that is the only place the headline error is taken.
• The shock is not what this case verifies. Its position on the three meshes is 4.550, 4.625 and 4.6625 m against Stoker’s 4.71535884 m at t = 1 s — 3.5 %, 1.9 % and 1.1 % short. It is approaching the right place under refinement, but a shock position resolved to a cell width on a 0.025 m mesh is a capture measurement, not an accuracy one, and it is reported here rather than claimed as agreement.
• Nothing is established about friction, a sloping bed or Coriolis. All three are absent from this case by construction, which is what makes the reference exact. A real dam break has all of them.