A well-balanced reconstruction for wetting/drying fronts

1. Introduction

• A well-balanced scheme prevents small steady-state perturbations from growing into numerical storms by exactly balancing flux and source terms and preserving lake-at-rest states to machine accuracy.The introduction motivates this requirement for coarse-grid simulations of quasi-steady solutions.Source: §1 · p. 1
• Dry areas pose a numerical challenge because the shallow-water system is not strictly hyperbolic when water height is zero, and oscillations can make the computed height negative.The system's characteristic speeds are u±√gh.Source: §1 · p. 1
• The new piecewise linear flow reconstruction for a continuous, piecewise linear bottom is designed to respect lake-at-rest and dry-lake states and preserve positive reconstructed water heights.It discretizes front cells to represent sloping shores and is also intended to handle combinations of these steady states and non-steady states appropriately.Source: §1 · p. 2
• The central-upwind scheme is positivity preserving with the reconstruction, but its proof may impose a severe time-step constraint; limiting outgoing fluxes when local draining time is shorter than the global step avoids reducing that step.The proof from carries over, and the flux-limiting technique is adopted from.Source: §1 · p. 2

2. A Central-Upwind Scheme for the Shallow Water Equations

• The semi-discrete scheme evolves finite-volume cell averages using central-upwind numerical fluxes and a source-term discretization, with the bottom represented by a continuous, piecewise linear approximation.The grid uses uniform finite-volume cells.Source: §2 · pp. 3–4
• The piecewise linear flow reconstruction uses a generalized minmod limiter for non-oscillatory behavior, and sets average velocity to zero when water height is below 10^-9 to avoid cancellation near dry areas.The threshold is used in all numerical experiments; the limiter parameter θ affects numerical viscosity.Source: §2 · pp. 4–5
• The correction algorithm in [12] restores positivity of the reconstruction but destroys the well-balancing property; the paper proposes an alternative positivity-preserving reconstruction that exactly preserves lake-at-rest and dry-lake steady states and their combinations.Source: §2 · p. 5
• Results from the second- and third-order SSP-RK solvers are almost undistinguishable in the paper’s examples.The numerical experiments use the third-order SSP-RK solver, with the timestep restricted by the standard CFL condition.Source: §2 · p. 5

• First-order piecewise-constant reconstruction can produce negative depths, while conventional piecewise-linear reconstruction does not guarantee positivity; the correction in preserves water amount but is unbalanced.Source: §3 · pp. 5–6
• The proposed wet/dry correction is designed to preserve positivity and well-balancing even in the presence of dry areas.Source: §3 · pp. 6–7
• They define a rest-state free-surface level by matching the area between it and the bottom to the cell's water volume, yielding trapezoidal or triangular wet geometry.Source: §3 · p. 7
• In partially flooded cells, the cell-average water level is not the free surface; a separate surface variable is defined only on the wet portion and remains above the bottom.Source: §3 · p. 8
• For an almost-dry cell beside a fully flooded one, they conserve its water while correcting the reconstruction; if one linear piece implies negative depth, they use two pieces.The wet/dry break point is determined by conservation.Source: §3 · pp. 8–9
• The authors note that one special case is non-generic and may arise only in under-resolved computations.Source: §3 · p. 10

Positivity Preserving and Well-Balancing

• Explicit Euler positivity can require a CFL-like restriction that no finite positive standard CFL condition (2.15) guarantees.In partially flooded cells, the associated timestep restriction can be serious.Source: §4 · pp. 11–12
• The draining-time technique from cuts off wet/dry-boundary fluxes only in cells at risk of emptying, enabling positivity with a uniform timestep under (2.15).The time-quadrature truncates space-time flux integrals at partially flooded interfaces.Source: §4 · pp. 11, 13
• With draining-time fluxes, nonnegative water heights remain nonnegative under the standard CFL condition (2.15).Theorem 4.1 establishes this for the water-height update (4.8).Source: §4 · p. 13
• The modified scheme preserves combinations of lake-at-rest and dry-lake steady states exactly, including in the presence of dry areas.Theorem 4.2 assumes constant free-surface values w_j and u=0 wherever h_j>0.Source: §4 · p. 14
• Replacing forward Euler with a higher-order SSP ODE solver preserves positivity without letting the local draining-time cutoff reduce or influence the global timestep.The global timestep is chosen at the first Runge–Kutta stage.Source: §4 · p. 15

5.1. Numerical accuracy order

• The accuracy test confirms that the proposed scheme is second-order accurate.The reported result uses a 12800-cell reference solution and is evaluated at t=0.1.Source: §5.1 · p. 17, Table 5.1 · p. 17

5.2. Still and oscillating lakes

• BCKN preserves the steady lake state in wet and dry regions, whereas KP introduces oscillations at the wet/dry boundary.The oscillations also influence solutions away from the boundary.Source: §5.2 · p. 17, Table 5.2 · p. 18, Figure 5.1 · p. 18
• Both KP and BCKN produce good oscillating-lake results with acceptable numerical order; BCKN discharge converges to the reference without oscillations.The BCKN solutions are compared at 200, 400, and 800 points against a 12800-point reference.Source: §5.2 · p. 18, Figure 5.3 · p. 19, Figure 5.2 · p. 19

5.3. Wave run-up on a sloping shore

• BCKN solutions decay to machine accuracy in the wave run-up test, while KP’s long-time convergence stalls near its truncation-error level.Both schemes give satisfactory solutions during the dynamic phase up to t=28.Source: §5.3 · p. 20, Figure 5.4 · p. 20, Figure 5.5 · p. 21

5.4. Dam-break over a plane

• For dam-break shocks entering dry areas, BCKN performs reasonably well despite not matching the resolution of the third- and fifth-order schemes in.The scheme is second-order accurate and was designed for well-balanced wet/dry equilibrium states.Source: §5.4 · p. 21, Figure 5.6 · p. 22

5.5. Laboratory dam-break over a triangular hump

• The triangular-hump dam-break predictions agree satisfactorily with measurements at all seven gauges and resolve wet/dry transitions sharply.The test includes bed friction; the scheme first updates without friction, then retains local acceleration from the bed-friction terms.Source: §5.5 · p. 23

6. Conclusion

• The wet/dry-front water-level reconstruction is conservative, well-balanced, and positivity preserving for both wet and dry cells.It is used with a second-order semi-discrete central-upwind scheme and continuous, piecewise linear bottom topography; positivity is ensured by cutting outflux across partially flooded edges when a cell runs empty.Source: §6 · p. 23
• Numerical examples demonstrate experimental order of convergence and well-balancing, while a predecessor scheme fails to converge to equilibrium.The new scheme is also reported as robust for shocks entering dry areas and simulations with Manning’s bottom friction, which is singular at wet/dry fronts.Source: §6 · pp. 23–24