6: Time Integration with RK4

[philipnickel]

Description

Implement explicit time integration using the classical 4th-order Runge-Kutta method. Create the full time-stepping loop that advances the free surface variables.

Background

The state vector y = [η, φ̃] evolves according to:

dy/dt = RHS(y)

where RHS involves:

1. Building K matrix from current η
2. Solving σ-Laplace for Φ
3. Recovering w̃
4. Evaluating free surface RHS

Each RK4 step requires 4 RHS evaluations (4 Laplace solves per time step).

Tasks

• Implement generic RK4 stepper
• Implement full RHS function
  • Input: current [η, φ̃]
  • Build K matrix
  • Assemble and solve Laplace
  • Recover w̃
  • Evaluate R_η, R_φ
  • Output: [R_η, R_φ]
• Implement time loop with output storage
• Implement CFL-based time step selection
• Test RK4 on simple ODE
• Implement standing wave test case

Acceptance Criteria

• RK4 achieves 4th-order convergence on test ODE (dy/dt = -y)
• Standing wave simulation runs for 10+ periods without crashing
• Oscillation period matches analytical T = 2π/ω within 1%
• Amplitude preserved within 5% after 10 periods
• Phase error < 10% after 10 periods

Test Configuration: Standing Wave

Domain: x ∈ [0, λ/2] where λ = 2π/k
BCs: Neumann (walls) at x = 0 and x = λ/2
Initial: η(x,0) = H·cos(kx), φ̃(x,0) = 0
Expected: η(x,t) = H·cos(kx)·cos(ωt)
Parameters: kh = 1, H/h = 0.01 (small amplitude)
Duration: 10 wave periods

Deliverables

• Time series plot of η at x = 0
• Comparison with analytical standing wave
• Energy vs time plot