GridConvergence

GridConvergence, in the context of numerical simulations, is the property that the computed solution approaches the exact solution of the governing equations as the computational grid is refined. It is a key aspect of code verification and mesh quality assessment, ensuring that numerical errors due to discretization decrease with mesh refinement rather than being dominated by modeling or solver issues.

Practically, a grid convergence study involves solving the problem on a sequence of grids with progressively smaller mesh spacings. For a quantity of interest, such as a surface pressure or a field value, one compares results across grids and analyzes how differences scale with grid spacing to estimate the converged value. Richardson extrapolation can be used to estimate the exact solution when the refinement ratio is known.

One formal metric is the grid convergence index (GCI), proposed by Roache, which provides a conservative estimate of remaining discretization error and a criterion for grid-independence. Using results from two or more grids and an assumed order of convergence, GCI yields a dimensionless measure of how close the finest solution is to the converged value.

Applications span computational fluid dynamics, structural analysis, acoustics, and electromagnetics. Best practices include using consistent physics and boundary conditions across grids, refining near steep gradients, and assessing grid quality through metrics such as cell aspect ratio and skewness. A grid-independent solution is typically declared when changes between the finest grids fall within a predefined tolerance.

Limitations include the possibility that a solution is not grid-convergent due to inadequate physical modeling, numerical diffusion, or mesh-induced effects. In some problems, full grid independence may be unattainable or impractical, requiring adaptive mesh refinement or alternative modeling approaches.