8: Stabilization for Steep Waves

[philipnickel]

Description

Implement stabilization techniques to enable simulation of steep, strongly nonlinear waves. The quartic nonlinear terms cause aliasing that leads to instability without proper treatment.

Background

From the paper: Neither over-integration alone nor filtering alone is sufficient. Both are required for steep waves (H/L ≈ 90% of breaking).

Over-integration: Evaluate nonlinear products on finer grid (2P) to prevent aliasing.

Spectral filtering: Gently damp highest modes to dissipate residual energy accumulation.

Tasks

• Over-integration implementation
  • Build LGL nodes for order 2P
  • Build interpolation matrix I: P+1 → 2P+1
  • Build L² projection matrix: 2P+1 → P+1
  • Modify RHS to compute products on fine grid
• Spectral filter implementation
  • Compute filter coefficients σ(i) with exponential rolloff
  • Build filter matrix F = V·diag(σ)·V⁻¹
  • Apply filter to η, φ̃ after each time step
• Stability testing with increasing wave steepness

Acceptance Criteria

• H/L = 10%: Stable without stabilization (baseline)
• H/L = 50%: Stable with stabilization
• H/L = 90%: Stable with both over-integration AND filter
• Stabilization does not degrade accuracy for mild waves (H/L = 10%)
• Cost overhead < 20% compared to unstabilized version

Filter Parameters (suggested starting point)

Cutoff: i_c = 0.9 * P
Strength: α = 0.05 (5% at highest mode)
Order: s = 1 (exponential)