3: Gradient Recovery for Vertical Velocity #13
Description
Description
Implement gradient recovery to compute the vertical velocity at the free surface, w̃ = ∂_z φ|_{z=η}. This is essential for evaluating the free surface evolution equations.
Background
In σ-coordinates, the vertical velocity is:
w̃ = ∂_z φ|_{z=η} = (1/d) · ∂_σ Φ|_{σ=1}
The gradient recovery extracts ∂_σ Φ at the surface and converts to physical coordinates.
Tasks
- Extract surface node indices (σ = 1)
- Apply differentiation matrix D_σ to compute ∂_σ Φ
- Extract surface values of ∂_σ Φ
- Compute w̃ = (1/d) · ∂_σ Φ|_{σ=1}
- Implement analytical w for Airy wave comparison
- Add validation test
Acceptance Criteria
- Constant Φ field gives w̃ = 0 (to machine precision)
- For Airy wave: w̃ matches analytical within 1% for P ≥ 6
- Error in w̃ decreases with increasing resolution
- Spectral convergence demonstrated
Optional Enhancement
- Implement L² projection smoothing if direct differentiation is noisy
- Solve: M · w = S · Φ for smooth w