Richardson Extrapolation Calculator
Enter coarse, medium, and fine results to estimate the order of convergence, the zero-step limit, the fine-result error, and the refinement needed for a target error.
Extrapolated limit1.5
- Observed or applied order p
- 2
- Estimated fine-result error
- 0.01
- Estimated relative error
- 0.33%
- Convergence pattern
- Monotonic converging
- Step size for target error
- 0.04
- Additional refinement factor
- 2.24
Refinement levels
| Level | Step size | Result |
|---|---|---|
| Coarse | 0.4 | 1.42 |
| Medium | 0.2 | 1.48 |
| Fine | 0.1 | 1.5 |
Assumes the leading discretization error behaves like C h^p over these three refinement levels.
How to use this calculator
- Enter the largest, intermediate, and smallest step sizes in strictly decreasing order.
- Enter the computed result at each step size for the same quantity of interest.
- Choose whether to estimate the observed order from the three results or use a known method order.
- Set the target absolute error to estimate how much smaller the next step must be.
How the extrapolation is calculated
Richardson extrapolation treats a numerical result as a limiting value plus a leading step-size error. The model is:
A(h) = A* + C h^p
- A(h) is the computed result at step size
h. - A* is the estimated result as the step size goes to zero.
- C is the leading error coefficient.
- p is the order of convergence.
The calculator labels the coarse, medium, and fine step sizes as h3 > h2 > h1, with results A3, A2, and A1. In estimated-order mode it solves the three-level equation:
(A3 - A2) / (A2 - A1) = (h3^p - h2^p) / (h2^p - h1^p)
When the refinement ratio is the same on both intervals, this reduces to p = ln(|(A3 - A2) / (A2 - A1)|) / ln(r). Unequal step sizes are handled by solving the equation numerically instead of forcing a halving assumption.
Once p is selected or estimated, the extrapolated limit is calculated from the fine and medium values:
A* = A1 + (A1 - A2) / ((h2 / h1)^p - 1)
The estimated fine-result error is |A* - A1|. The relative error is that absolute error divided by |A*|, when the extrapolated limit is not zero. For a target absolute error T, the projected step is h_target = h1 (T / e1)^(1 / p). If the current fine result already meets the target, the target step is capped at the current fine step and the refinement factor is reported as 1.
What moves the result most
The order p has the strongest effect on both the limit and the projected next step. A second-order method cuts the leading error by about four when the step is halved, while a first-order method cuts it by about two. The pattern of the three results also matters. Shrinking differences with the same sign are the cleanest case; opposite signs can happen, but they are a warning that the leading-error model may not describe the data well.
What this calculator leaves out
This tool does not know whether the underlying simulation, mesh, finite-difference stencil, quadrature rule, or time integrator is appropriate. It also does not include iterative solver error, modeling error, measurement error, mesh-quality changes, or roundoff beyond flagging nearly identical results. Treat the target step as a planning estimate, then confirm it with another refined calculation.
Worked example
Suppose a calculation gives A3 = 1.42 at h3 = 0.4, A2 = 1.48 at h2 = 0.2, and A1 = 1.495 at h1 = 0.1. The successive differences are -0.06 and -0.015, so with a constant refinement ratio of 2 the observed order is ln(4) / ln(2) = 2.
The extrapolated limit is 1.495 + (1.495 - 1.48) / (2^2 - 1) = 1.5. The estimated fine-result error is |1.5 - 1.495| = 0.005, or about 0.33% of the extrapolated limit. For a target absolute error of 0.001, the projected step is 0.1 (0.001 / 0.005)^(1 / 2) = 0.0447, so the additional refinement factor is about 2.24.
Common questions
What is Richardson extrapolation?
Richardson extrapolation estimates the zero-step or continuum value of a numerical calculation by assuming the leading error follows a power law in the step size. With enough refined results, it can estimate both the limiting value and the error remaining in the fine result.
How do I estimate observed order of convergence?
Use three results for the same quantity at progressively smaller step sizes. The calculator compares the two successive result differences and solves for the order that makes those differences consistent with a leading C h^p error term.
Do the step sizes need to be halved each time?
No. Equal halving is convenient, but the calculation can use unequal positive step sizes as long as they are strictly ordered from coarse to fine. Ratios very close to 1 make the order estimate sensitive to roundoff and other errors.
What is the difference between theoretical and observed order?
The theoretical order comes from the numerical method under ideal assumptions. The observed order is measured from actual computed results, so boundaries, meshes, nonlinear solvers, time stepping, and roundoff can make it lower or unstable.
What does oscillatory convergence mean?
Oscillatory convergence means the result crosses around the apparent limit instead of approaching it from one side. It can still shrink toward a stable value, but the simple leading-power extrapolation is less secure and should be checked with another refinement.
Why can Richardson extrapolation fail near machine precision?
Near machine precision, the differences between successive results can be dominated by roundoff rather than discretization error. When that happens, the computed order may be indeterminate or misleading even though the numerical results appear very close together.