Grid Convergence Index (GCI) Guidelines

Grid Convergence Index (GCI) Guidelines

Overview

The Grid Convergence Index (GCI) is a standardized method for reporting discretization uncertainty in CFD and numerical simulations. It was developed by Patrick Roache and is recommended by ASME V&V 20 and AIAA standards.

Three-Grid GCI Procedure

Requirements

• Three systematically refined grids with spacings h1 < h2 < h3
• Constant or near-constant refinement ratios r21 = h2/h1, r32 = h3/h2
• Recommended: r >= 1.3 (to ensure measurable differences)
• Monotone convergence (no oscillation in solution values)

Step-by-Step Calculation

1. Compute refinement ratios:

   r21 = h2 / h1
   r32 = h3 / h2
   

2. Compute observed order p:

   p = |ln|e32/e21|| / ln(r21)
   

   where e32 = f3 - f2 and e21 = f2 - f1.

   For non-uniform ratios, an iterative procedure using:

   p = |ln|e32/e21| + ln((r21^p - s) / (r32^p - s))| / ln(r21)
   

   where s = sign(e32/e21). This reduces to the simple formula when r21 = r32.

3. Compute GCI for fine grid:

   GCI_fine = Fs * |e21/f1| / (r21^p - 1)
   

   where Fs is the safety factor.

4. Compute GCI for coarse grid:

   GCI_coarse = Fs * |e32/f2| / (r32^p - 1)
   

5. Check asymptotic ratio:

   AR = GCI_coarse / (r21^p * GCI_fine)
   

   If AR is approximately 1.0 (within 10%), the grids are in the asymptotic range and the GCI is reliable.

6. Richardson extrapolated value:

   f_extrap = f1 + (f1 - f2) / (r21^p - 1)
   

Safety Factors

The factor of 1.25 is analogous to a 95% confidence interval for well-behaved convergence data.

ASME V&V 20 Standard

The ASME V&V 20 standard (Verification and Validation in Computational Fluid Dynamics and Heat Transfer) recommends:

• Using at least 3 systematically refined grids
• Reporting GCI with the fine-grid solution
• Checking the asymptotic ratio
• Documenting the observed convergence order
• Reporting the Richardson-extrapolated value as the best estimate

Practical Guidelines

Grid Design

• Use constant refinement ratio across all directions
• Recommended ratio: r = 1.5 to 2.0
• Avoid r < 1.3 (differences may be in round-off noise)
• Avoid r > 3.0 (large jumps may skip pre-asymptotic behavior)

Common Issues

1. Observed order much higher than expected: Possible superconvergence or error cancellation. Verify with more grid levels.

2. Observed order much lower than expected: Solution may not be in the asymptotic range. Use finer grids.

3. Negative observed order: Solution is diverging. Check for coding errors, boundary condition issues, or inadequate resolution.

4. Oscillatory convergence: The GCI method does not apply. Consider using bounding approaches or the range of solutions as the uncertainty.

5. Very small GCI (< 0.1%): Solution may be grid-independent already, or the quantity of interest is insensitive to grid refinement.

Reporting

When reporting GCI results, include:

• The three grid spacings and corresponding solution values
• The observed convergence order
• The GCI value for the fine grid (as a percentage)
• The Richardson-extrapolated value
• The asymptotic ratio
• Whether the solution is in the asymptotic range

Example Report Format

Grid spacings: h1=0.01, h2=0.02, h3=0.04
Refinement ratio: r = 2.0
Solution values: f1=1.0008, f2=1.0032, f3=1.0128
Observed order: p = 2.00
GCI_fine = 0.027%
Richardson extrapolated value: 1.00000
Asymptotic ratio: 1.000
Conclusion: Solution is in asymptotic range; GCI is reliable.