FIGURE 1.

Convergence of the level set algorithms. (a) Convergence of the second-order interface locating algorithm by Chopp (17). A signed distance function, defined in a band of half-width 3h around a sphere, is reinitialized. The relative L₂ (squares) and L_∞ (circles) errors of the present implementation (solid symbols) are compared to the numbers published by Chopp (17) (open symbols) and the theoretical second-order scaling (solid line). The errors are computed from all points immediately adjacent to the interface. (b) Convergence of the second-order orthogonal extension. The spherical harmonic (Eq. 11) is initialized in a band of half-width 3h and orthogonally extended outwards to a 9h band. This is done over the exact signed distance function ψ, initialized in a 12h band (centered differences are used for ∇ψ in the extension). To compute the error, ∇u is computed using fourth-order centered differences in the band 3h < |ψ| < 6h, and ∇ψ is known analytically. The error is computed and the absolute L₂ (squares) and L_∞ (circles) norms are reported.