5: Free Surface RHS Evaluation

[philipnickel]

Description

Implement evaluation of the right-hand side of the Zakharov free surface evolution equations. These are the kinematic and dynamic boundary conditions that govern wave evolution.

Background

The Zakharov formulation evolves surface quantities η and φ̃:

Kinematic (mass conservation):

∂η/∂t = -∂ₓη · ∂ₓφ̃ + w̃·(1 + (∂ₓη)²)

Dynamic (momentum/Bernoulli):

∂φ̃/∂t = -gη - (1/2)·[(∂ₓφ̃)² - w̃²·(1 + (∂ₓη)²)]

Note: Contains up to quartic nonlinear terms (w̃² · (∂ₓη)²).

Tasks

• Implement surface differentiation (1D) for ∂ₓη, ∂ₓφ̃
• Implement kinematic RHS: R_η
• Implement dynamic RHS: R_φ
• Test with still water (should give zero)
• Test with Airy wave (compare to analytical ∂ₜη, ∂ₜφ̃)

Acceptance Criteria

• Still water (η=0, φ̃=0, w̃=0): R_η = 0, R_φ = 0 exactly
• Airy wave: R_η matches analytical ∂ₜη within 1%
• Airy wave: R_φ matches analytical ∂ₜφ̃ within 1%
• Nonlinear terms computed correctly (verify individual terms)

Technical Notes

• All operations are pointwise at surface nodes
• Need to be careful with the sign conventions
• The quartic terms will cause aliasing issues for steep waves (addressed in Issue Code environment stuff before handin #8)