Глибинні рівняння води: руйнування дам, бори та розв’язок Roe

Collapse of 3D flow into two equations

The shallow water equations (also called the Saint-Venant equations) describe a thin layer of fluid whose horizontal extent vastly exceeds its depth — a river, a flooded plain, a tsunami crossing an ocean basin, or the atmosphere's large-scale weather systems. Depth-averaging the full 3D incompressible Navier-Stokes equations under a hydrostatic pressure assumption collapses the whole vertical structure into two conserved quantities per point: depth h and depth-averaged momentum hu.

∂U/∂t + ∂F(U)/∂x = S U = [h, hu]ᵀ, F(U) = [hu, hu² + ½gh²]ᵀ Wave speeds (eigenvalues of the flux Jacobian): λ = u ± c, c = √(gh) Froude number: Fr = |u| / c → Fr1 supercritical The celerity c = √(gh) plays the same role as the speed of sound in gas dynamics, and Fr is the direct shallow-water analogue of the Mach number — which is why the same numerical machinery built for compressible aerodynamics carries over almost unchanged.

∂U/∂t + ∂F(U)/∂x = S      U = [h, hu]ᵀ,  F(U) = [hu, hu² + ½gh²]ᵀ
Wave speeds (eigenvalues of the flux Jacobian): λ = u ± c,  c = √(gh)
Froude number: Fr = |u| / c   →  Fr<1 subcritical, Fr>1 supercritical

Roe's approximate Riemann solver

A finite-volume scheme updates each cell from the fluxes at its two interfaces. Computing the flux at an interface between two different states — a "Riemann problem" — has an exact solution, but it is nonlinear and iterative. Philip Roe's 1981 trick is to linearise it: build a specially weighted average state (û, ĉ) from the left and right cells, and solve the resulting linear problem exactly.

û = (√h_L·u_L + √h_R·u_R) / (√h_L + √h_R) Roe-averaged velocity

ĉ = √( g·(h_L + h_R)/2 ) Roe-averaged celerity

F_i+½ = ½(F_L + F_R) − ½ Σ |λ̂ₖ| αₖ rₖ the |λ̂| term is numerical diffusion

That diffusion term is what keeps the scheme stable across a shock — it smears a discontinuity over a few cells instead of oscillating around it — but on its own it is only first-order accurate. MUSCL reconstruction (Monotone Upstream-Centred Schemes for Conservation Laws) recovers second order by extrapolating a limited linear profile within each cell before the Roe solve, using a slope limiter — minmod, van Leer, or superbee — that suppresses the spurious oscillations a naive linear extrapolation would create near a shock.

û = (√h_L·u_L + √h_R·u_R) / (√h_L + √h_R)     Roe-averaged velocity
ĉ = √( g·(h_L + h_R)/2 )                      Roe-averaged celerity
F_i+½ = ½(F_L + F_R) − ½ Σ |λ̂ₖ| αₖ rₖ          the |λ̂| term is numerical diffusion

The CFL condition: how far one step is allowed to travel

Because the scheme is explicit, the timestep cannot outrun the fastest wave in the domain. The Courant-Friedrichs-Lewy condition caps Δt so that information never crosses more than one cell per step:

Δt ≤ CFL · Δx / max(|u| + c)     safe CFL ≈ 0.9 first-order, ≈ 0.5 with MUSCL+RK2

The dry-bed problem

Flood simulations routinely have to deal with dry ground, where h → 0. That is dangerous numerically: the celerity c = √(gh) vanishes, velocity u = hu/h becomes a division near zero, and a naive Roe solver can produce negative depths and diverge outright.

Practical fixes include a depth threshold that treats near-zero cells as solid walls, a positivity-preserving limiter that clips h ≥ 0 after every step, and — the more rigorous option — hydrostatic reconstruction (Audusse et al., 2004), which reconstructs interface depths from the bed slope so that they are guaranteed non-negative by construction. That same reconstruction also gives the scheme the well-balanced property: a lake at rest with a bumpy bed stays exactly at rest, rather than drifting under numerical noise from imperfectly cancelled flux and source terms.

One equation, many phenomena

The same two-equation system, with different source terms switched on, reproduces a striking range of real flows. A sudden dam-break is almost pure inertia and pressure, splitting into a rarefaction wave racing upstream and a shock-like bore racing downstream — the textbook Riemann problem this article's simulation solves directly. A tidal bore adds bed friction to that same balance. An ocean tsunami is dominated by pressure over enormous length scales, travelling near 200 m/s in deep water. Add the Coriolis term f = 2Ω sin(φ) and the equations describe geostrophic ocean eddies and storm surges, where the Earth's rotation balances the pressure gradient over scales of a hundred kilometres or more.

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