Quantifies the discretization error of a CFD result from solutions on three systematically refined meshes, using Richardson extrapolation. It reports the observed order of accuracy, the extrapolated zero-grid-spacing value and the Grid Convergence Index (GCI).
Inputs
One representative solution quantity (drag, Nusselt number, pressure drop, outlet temperature) on three meshes, plus the mesh sizes. Defaults are a clean second-order demo on a 1 m square.
Solution quantity on three meshes
Solution on the finest mesh.
Solution on the medium mesh.
Solution on the coarsest mesh.
Mesh description
Switches the representative cell size formula.
X-direction extent of the domain.
Y-direction extent of the domain.
Z-direction extent, 3D only.
Cell counts, fine mesh
Cell counts, medium mesh
Cell counts, coarse mesh
Results
Grid Convergence Index GCI21
-
Estimated discretization uncertainty band on the fine-mesh result, safety factor 1.25.
Observed order p
-
Compare with the formal order of your discretization scheme.
Extrapolated value φext
-
Richardson estimate at zero grid spacing.
Convergence mode
-
Representative cell sizes h1, h2, h3
-
Fine, medium, coarse. Units of m.
Refinement ratios r21, r32
-
Each level should differ by at least 30 percent for a reliable study.
Approximate relative error ea21
-
Extrapolated relative error eext21
-
How it works
The calculator follows the standard three-grid procedure for estimating discretization error in CFD. It assumes the three meshes are systematically refined versions of each other and that the solutions are in the asymptotic range, where the error shrinks as a power of the cell size.
Compute a representative cell size for each mesh: the square root of (domain area / cell count) in 2D, the cube root of (domain volume / cell count) in 3D.
Form the refinement ratios r21 = h2/h1 and r32 = h3/h2, and the solution differences ε21 = φ2 - φ1 and ε32 = φ3 - φ2.
Check the sign of ε32/ε21: a negative sign means oscillatory convergence, a positive sign means monotonic convergence.
Solve for the observed order of accuracy p from the transcendental relation between the refinement ratios and the solution differences, using bisection.
Extrapolate the solution to zero grid spacing with the Richardson formula.
Report the approximate relative error, the extrapolated relative error and the Grid Convergence Index with the recommended safety factor of 1.25 for three-grid studies.
Solution quantity on the fine, medium and coarse meshes
h1, h2, h3
Representative cell sizes of the three meshes, m
r21, r32
Grid refinement ratios between consecutive mesh levels
ε21, ε32
Solution differences between consecutive mesh levels
p
Observed order of accuracy of the discretization
s
Sign indicator: +1 for monotonic, -1 for oscillatory convergence
φext
Richardson-extrapolated solution at zero grid spacing
ea21, eext21
Approximate and extrapolated relative errors
GCI21
Grid Convergence Index on the fine mesh, an uncertainty band
Fs
Safety factor, 1.25 when three meshes are used
References
Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H. and Raad, P. E., Procedure for estimation and reporting of uncertainty due to discretization in CFD applications, Journal of Fluids Engineering, 130(7), 2008.
Richardson, L. F., The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philosophical Transactions of the Royal Society A, 210, 1911.
Roache, P. J., Verification and Validation in Computational Science and Engineering, Hermosa Publishers, 1998.
