Mesh Convergence Studies — GCI & Richardson Extrapolation

How do I perform a proper mesh convergence study?

Generate at least three systematically refined meshes with a consistent refinement ratio (typically 1.5-2x). Extract the key output quantity from each mesh. Compute the Grid Convergence Index (GCI) — most standards (NASA, ASME) recommend GCI < 5% for publishable results. Richardson extrapolation estimates the exact solution from the refinement trend.

What is Richardson extrapolation and what does it tell you?

Richardson extrapolation: phi_exact ~ phi_fine + (phi_fine - phi_coarse)/(r^p - 1), where r is the refinement ratio and p is the observed order of convergence. The observed order p = log((phi_coarse - phi_medium)/(phi_medium - phi_fine))/log(r). If p matches the theoretical order of your scheme, you are in the asymptotic convergence range. If p is far from theoretical, mesh-dependent features or singularities need attention.