3: Gradient Recovery for Vertical Velocity

[philipnickel]

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