Richardson Extrapolation Calculator for CFD Mesh Refinement

Estimate an extrapolated grid-independent value from three CFD mesh-refinement results and review apparent convergence behavior for numerical verification.

PublishedCalculatorengineering calculators

Input Parameters

Solution Values

Refinement Ratios

Must be > 1.0 (e.g. 2.0 for doubling elements in each direction).

Known Order (Optional)

If left blank, the calculator will attempt to estimate the apparent order from the three solutions.

Load example data:

Worked Example

Scenario: Pressure Drop Coefficient

You run a CFD simulation on 3 progressively refined grids to compute the Pressure Drop Coefficient or a proxy value. You need to estimate the true grid-independent value and assess apparent convergence.

Example Inputs:
  • Coarse-grid result (φ₃): 1.842
  • Medium-grid result (φ₂): 1.791
  • Fine-grid result (φ₁): 1.775
  • Coarse-to-medium ratio (r₃₂): 2.0
  • Medium-to-fine ratio (r₂₁): 2.0
  • Known Order p: (left blank)

Interpretation: The result decreases from coarse to medium to fine, so the sequence is monotonic. With consistent refinement ratios, the calculator estimates an apparent order of convergence and uses Richardson extrapolation to estimate a grid-independent value. The extrapolated value should be interpreted as a numerical verification estimate for this selected quantity of interest, not as validation of the physical model.

How to use this calculator

Richardson extrapolation estimates a grid-independent value from a sequence of refined CFD solutions. The result is only meaningful when the grids form a systematic refinement sequence and the selected quantity of interest shows a reasonable convergence trend.

If the theoretical order of your numerical scheme is known (e.g., exactly 2.0 for a pure second-order scheme), you can input it manually. Otherwise, leave it blank to let the calculator estimate the apparent order from the three solutions.

The Formulas

ϕext=ϕ1+ϕ1−ϕ2r21p−1\phi_{ext} = \phi_1 + \frac{\phi_1 - \phi_2}{r_{21}^p - 1}

Where ϕ1\phi_1 is the fine grid result, ϕ2\phi_2 is the medium grid result,r21r_{21} is the refinement ratio, and pp is the apparent order of accuracy.

Limitations

    • Requires Monotonic Convergence: If the solution oscillates or diverges between grids, Richardson extrapolation is invalid and the calculator will show a warning.
    • Refinement Ratios: Ratios should ideally be greater than 1.1 to separate discretization error from round-off noise.
    • Does Not Validate Physics: Richardson extrapolation supports numerical verification. It does not validate the physical model, boundary conditions, turbulence model, or wall treatment.