Level Set Method

Advecting φ with velocity preserves the zero level set (interface), but the gradient |∇φ| drifts from 1. Over time, φ becomes steep or flat, causing numerical errors. Reinitialization solves a PDE (e.g., distance redistancing) to restore |∇φ| = 1 while preserving the zero level set. Without it, curvature calculations become inaccurate and interfaces thicken. Trade-off: reinitialization fixes distance but introduces diffusion and mass loss.

Curvature is κ = ∇·(∇φ/|∇φ|), the divergence of the unit normal. Since n̂ = ∇φ/|∇φ|, you compute gradients of φ, normalize, then take divergence. Finite-difference approximations introduce noise, especially near the interface where |∇φ| ≈ 0. Smoothing φ locally before computing κ reduces oscillations. Accurate curvature is critical for surface tension forces.

VOF is inherently conservative: volume fractions integrate to phase volume exactly. Level sets do not conserve mass; reinitialization dissipates interface region and causes volume loss (~1-2% per reinit in 2D, worse in 3D). Hybrid VOF-LS methods or mass-correction algorithms (recomputing velocity or pressure to conserve volume) address this. For engineering applications requiring strict conservation, VOF is preferred.

No explicit interface points means no merging/splitting logic needed. Complex topology changes happen naturally. Interface curvature and geometry are easily computed from φ gradients. No need for interface parameterization. Drawback: implicit representation is less intuitive, and level sets have weaker mass conservation.

Use a Fast Marching Method (FMM) or Fast Sweeping Method (FSM) to compute the signed distance from the initial interface. For a simple interface (circle, ellipse), an analytical formula works. For complex initial geometries (e.g., from segmented images), FSM is efficient and accurate. Ensure |∇φ| = 1 at initialization to avoid numerical errors in early time steps.