Choosing a Flow Algorithm

ANUGA’s flow algorithm controls how the shallow water equations are advanced in time. All available algorithms use the same 2nd-order discontinuous-elevation spatial reconstruction (the “DE” family), but differ in their time-stepping scheme and the aggressiveness of their slope limiters.

domain.set_flow_algorithm('DE1')   # set before evolve

Quick comparison

Algorithm	Time order	Flux calls / step	CFL	Slope limiter β	When to use
DE0	1st	1	0.9	0.5 (conservative)	Fast exploratory runs; very large or very dry domains
DE1	2nd	2	1.0	1.0 (full)	General-purpose default — robust and accurate
DE_ader2	2nd	1	1.0	1.0 (full)	Same accuracy as DE1 at the cost of DE0 — recommended for production
DE2	3rd	3	1.0	1.0 (full)	Rarely needed; reserved for highly transient flows

Recommendation

• Start with DE1. It is 2nd-order in both space and time, uses the full slope limiter, and is the most widely validated algorithm in ANUGA. It is roughly half the speed of DE0 (two flux evaluations per step), but produces materially better results in flows with steep gradients, bores, or rapidly varying inundation fronts.

• Use DE0 when speed matters more than temporal accuracy. It is approximately twice as fast as DE1 per simulated second, but uses a more conservative slope limiter (β = 0.5) and first-order time stepping. Results are still 2nd-order in space, so spatial gradients are well resolved; the 1st-order time stepping introduces a mild diffusivity that is often acceptable for long-duration flood studies on large, mildly varying domains.

• DE_ader2 is the future default. It achieves the same 2nd-order temporal accuracy as DE1 but requires only one flux call per timestep (the same as DE0) by using an ADER Cauchy–Kovalewski predictor. The first timestep bootstraps with a plain Euler step to establish the initial CFL dt; from the second step onwards it runs at full 2nd-order accuracy. It is still being validated across the full range of ANUGA’s test cases, so DE1 remains the safer choice for critical production work at present.

• Avoid DE2 in routine use. The SSP-RK3 scheme adds a third flux call per step (50% more expensive than DE1) for only marginal accuracy gains in the smooth-solution regime typical of shallow water problems. Reserve it for specialised studies where 3rd-order temporal accuracy is demonstrably needed.

Algorithm details

DE0 — Euler, CFL 0.9

domain.set_flow_algorithm('DE0')

• Time stepping: forward Euler — one extrapolation and one flux call per step.

• CFL factor: 0.9 (10% safety margin; slightly more conservative than the other algorithms).

• Slope limiter: β = 0.5, applied to both wet and transitional cells. The reduced β damps spurious oscillations near wet/dry fronts and makes DE0 more robust on very heterogeneous terrain.

• Minimum allowed height: 1 × 10⁻¹² m (effectively zero); water can persist in very thin layers.

• Cost: ~1× (baseline).

DE1 — Runge–Kutta 2, CFL 1.0

domain.set_flow_algorithm('DE1')

• Time stepping: 2nd-order Runge–Kutta (Heun’s predictor–corrector) — two flux calls per step.

• CFL factor: 1.0.

• Slope limiter: β = 1.0 (full reconstruction, no artificial limiting in smooth regions).

• Minimum allowed height: 1 × 10⁻⁵ m.

• Cost: ~2× relative to DE0.

DE_ader2 — ADER-2 Cauchy–Kovalewski, CFL 1.0

domain.set_flow_algorithm('DE_ader2')

• Time stepping: 2nd-order ADER scheme using a Cauchy–Kovalewski (C-K) predictor. After an initial Euler bootstrap step, each subsequent step requires only a single flux call: edge values are advanced to the midpoint time level Q^n+1/2 in-place using local SWE derivatives, and the flux is computed from those predicted edges.

• CFL factor: 1.0.

• Slope limiter: β = 1.0 (same as DE1).

• Minimum allowed height: 1 × 10⁻⁵ m (same as DE1).

• Cost: ~1× (same as DE0 after the bootstrap step).

• Note: The predictor reuses the timestep from the previous step (_ader2_prev_dt), which is the global-minimum CFL dt across all MPI ranks. This ensures consistency in parallel runs.

DE2 — SSP-RK3, CFL 1.0

domain.set_flow_algorithm('DE2')

• Time stepping: strong-stability-preserving 3rd-order Runge–Kutta — three flux calls per step.

• CFL factor: 1.0.

• Slope limiter: β = 1.0.

• Minimum allowed height: 1 × 10⁻⁵ m.

• Cost: ~3× relative to DE0.

Algorithm and GPU compatibility

All four algorithms run on both CPU and GPU (OpenMP target offloading). See GPU Acceleration (OpenMP target offloading) for a full compatibility table and instructions on enabling GPU acceleration.

Low-Froude correction

The standard Kurganov–Noelle–Petrova (KNP) central-upwind flux scheme applies a numerical diffusion term proportional to the local wave speed. In subcritical or nearly-still-water flows — estuaries, tidal flats, ponded floodplains — the velocity is much smaller than the wave speed (\(\mathrm{Fr} = |\mathbf{u}| / \sqrt{gh} \ll 1\)) and this diffusion can damp slow-moving features more than the physical solution warrants.

The low_froude setting reduces the momentum diffusion term by a factor local_fr computed from the local Froude number at each edge:

domain.set_low_froude(0)   # default — no correction
domain.set_low_froude(1)   # aggressive clamping for strongly subcritical flows
domain.set_low_froude(2)   # smooth scaling, recommended for tidal / estuary

Value	Name	Effect on momentum diffusion scaling factor local_fr
0	LOW_FROUDE_OFF	local_fr = 1.0 — no correction; standard KNP diffusion
1	LOW_FROUDE_1	local_fr = sqrt(clamp(Fr², 0.001, 1.0)) — clamps between ~0.032 and 1.0; aggressive reduction for strongly subcritical flows
2	LOW_FROUDE_2	local_fr = sqrt(clamp(Fr², 0.01, 1.0)) with smooth floor — stays at 0.1 for Fr < 0.01 then ramps smoothly to 1.0; gentler and recommended for mixed-regime problems

Only the momentum diffusion terms are scaled; the mass flux is never modified, so conservation is preserved exactly.

When to use it

• 0 (default): suitable for most coastal, flood, and dam-break problems where the flow regularly passes through or near critical (Fr ≈ 1).

• 1: strongly subcritical flows where Fr stays well below 0.1 throughout (e.g., estuarine hydraulics, slow tidal inundation). The aggressive clamping provides maximum diffusion reduction but can introduce mild oscillations near steep gradients.

• 2: the recommended non-zero choice for problems with mixed flow regimes (some fast, some very slow reaches). The smoother ramp avoids the step-change behaviour of mode 1 near Fr ≈ 0.001.

The setting is independent of the flow algorithm and can be combined with any of DE0, DE1, DE_ader2, or DE2:

domain.set_flow_algorithm('DE1')
domain.set_low_froude(2)

Reference

Domain.set_flow_algorithm(algorithm: str = 'DE0') → None

Set combination of slope limiting and time stepping

Currently

DE0 DE1 DE2 DE0_7 DE1_7

Domain.get_flow_algorithm() → str

Get method used for timestepping and spatial discretisation

Domain.set_low_froude(low_froude: int = 0) → None

For low Froude problems the standard flux calculations can lead to excessive damping. Set low_froude to 1 or 2 for flux calculations which minimize the damping in this case.