Results of the SPH evaluation of the test profiles A, B, and C using different VEs. Top row: density profiles of models A (Gaussian), B (Inverted-Gaussian), and C (Wall), of Table 1, calculated with X₁ using p = 0.0 (magenta) and p = 0.8 (blue) in Eq. (13). Crosses (×) in black are for X₂ (Eq. (15), with p = 1) and points in light blue are the analytic values. Central row: E₁ (partition of unity) error for the same three profiles following the same color schemes, with the black lines for X₂ with p = 1. Bottom row: same as central row but for error E₂ = |⟨Δx⟩|/h.