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Grid Refinement Richardson Single Term

Calculate one Richardson fine-grid error and extrapolation under an explicit asymptotic h^p model and positive supplied order.

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Description

Calculate a supplied Richardson fine-grid error term and extrapolated value.

Numerical physics calculations depend on discretization, index conventions, boundary conditions and model units. A local force kick is only part of an integration method, a stencil residual is not a solved field, and a one-sided power weight is not a spectral-density normalization. Floating-point resolution also limits grid and coordinate representations.

Calculate a supplied Richardson fine-grid error term and extrapolated value.

When to use this tool

Use Grid Refinement Richardson Single Term when you already have coarseValue, fineValue, refinementRatio, convergenceOrder and need to calculate a supplied Richardson fine-grid error term and extrapolated value.

How to use the tool and how it works

  1. Enter coarseValue, fineValue, refinementRatio, convergenceOrder. Use the units and conventions below.
  2. Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
  3. Read estimatedFineError, extrapolatedValue. Use the output convention below.

Require refinementRatio>1 and convergenceOrder positive/finite; denominator=expm1(order*ln(ratio)) must be finite and positive. correction=(fine-coarse)/denominator; return estimatedFineError=-correction and extrapolated=fine+correction.1

Richardson extrapolation relies on an assumed asymptotic error order across comparable grids. Two supplied values and a chosen order do not demonstrate that this convergence regime has been reached.2

Input fieldMeaning and unitsAccepted input
coarseValueCoarse ValueRequired. Finite number.
fineValueFine ValueRequired. Finite number.
refinementRatioRefinement RatioRequired. Finite number; strictly above 1.
convergenceOrderConvergence OrderRequired. Finite number; strictly above 0.
Output fieldInterpretationReturned shape
estimatedFineError(coarseValue-fineValue)/(refinementRatio^convergenceOrder-1), opposite to the extrapolation correction.Required. Finite number.
extrapolatedValuefineValue+(fineValue-coarseValue)/(refinementRatio^convergenceOrder-1).Required. Finite number.
json
{
  "input": {
    "coarseValue": 2,
    "fineValue": 1.25,
    "refinementRatio": 2,
    "convergenceOrder": 2
  },
  "output": {
    "estimatedFineError": 0.25,
    "extrapolatedValue": 1
  }
}

Limitations and assumptions

  • Error sign is fine-minus-extrapolated. The order and asymptotic power-law regime are assumptions, not verified convergence; no solver or complete error bound is established.
  • Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
  • Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.

Alternative or Complementary methods

Use the complete discretized model, integration algorithm and boundary treatment. Assess convergence, conservation and stability with the full calculation; individual residuals, CFL components, relaxation terms and index mappings do not establish numerical accuracy or physical validity.

References

  1. Calculation and input contract: computational-physics/grid-refinement-richardson-single-term — tool.io

  2. Examining Spatial (Grid) Convergence — NASA Glenn Research Center

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