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Estimating Uncertainties in Statistics Computed from DNS

Todd A. Oliver, Nicholas Malaya, Rhys Ulerich, Robert D. Moser

arXiv:1311.0828v1physics.flu-dynnlin.CDphysics.comp-ph

TL;DR

DNS statistics require uncertainty estimates that account for both finite sampling and discretization, but correlated sampling makes standard Richardson extrapolation unreliable. The paper develops a correlation-aware sampling estimator and a Bayesian Richardson extension, finding generally small discretization errors while showing that sampling error does not always dominate.

  • Problem

    Correlated finite sampling and discretization both contribute uncertainty to DNS statistics, limiting the reliability of standard discretization-error estimates.

  • Method

    The paper combines a sampling-error estimator that accounts for correlation with a Bayesian extension of Richardson extrapolation.

  • Results

    Sampling uncertainty can make the order of accuracy difficult to determine, while estimated discretization errors are generally small for the tested DNS quantities.

  • Takeaways & Limitations

    Usual DNS resolution heuristics appear adequate for many quantities, but discretization error should be estimated because it can be comparable to or larger than sampling error.

  • Takeaways & Limitations

    The paper notes that practical DNS discretization errors may not be in the asymptotic regime.

Abstract

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Rigorous assessment of uncertainty is crucial to the utility of DNS results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier-Stokes equations. Due to the presence of non-trivial sampling error, standard techniques for estimating discretization error (such as Richardson extrapolation) fail or are unreliable. This work provides a systematic and unified approach for estimating these errors. First, a sampling error estimator that accounts for correlation in the input data is developed. Then, this sampling error estimate is used as part of a Bayesian extension of Richardson extrapolation in order to characterize the discretization error. These methods are tested using the Lorenz equations and are shown to perform well. These techniques are then used to investigate the sampling and discretization errors in the DNS of a wall-bounded turbulent flow. For both cases, it is found that while the sampling uncertainty is large enough to make the order of accuracy difficult to determine, the estimated discretization errors are quite small. This indicates that the commonly used heuristics provide ad- equate resolution for this class of problems. However, it is also found that, for some quantities, the discretization error is not small relative to sampling error, indicating that the conventional wisdom that sampling error dominates discretization error for this class of simulations needs to be reevaluated.

I. INTRODUCTION

DNS statistics contain both sampling and discretization uncertainty, making standard discretization-error estimates unreliable. This work develops correlation-aware sampling estimates and a Bayesian Richardson approach, then applies them to Lorenz equations and turbulent channel-flow DNS.

  • The paper develops a systematic method for estimating uncertainty in statistics computed from DNS, addressing both sampling and discretization error.
  • Standard Richardson extrapolation is unreliable for DNS because sampling uncertainty can strongly affect the estimated discretization error.
  • Sampling error contaminates DNS statistics because time-history and spatial samples are generally correlated rather than independent.Treating widely separated snapshots as independent can underestimate uncertainty when separation is insufficient and overestimate it when separation is excessive.
  • The sampling estimator fits autoregressive models to estimate autocorrelation, while the Bayesian Richardson extension incorporates statistical uncertainty and prior accuracy information.The Bayesian formulation reduces the sensitivity of discretization-error estimates to finite-sampling effects.
  • In turbulent channel-flow DNS, discretization errors were generally much less than one percent for several quantities, suggesting usual mesh heuristics are adequate.For some other quantities, the model was invalidated and no confident discretization-error estimates were presented.
  • For some quantities, discretization error was similar to or larger than sampling error, challenging the assumption that sampling error always dominates.

II. METHODOLOGY

The methodology separates finite sampling and discretization uncertainty, estimating correlated-data sampling error and using it in a Bayesian Richardson framework. The estimators are assessed on Lorenz-equation simulations.

  • The work estimates finite sampling and discretization errors as the two major uncertainty sources in DNS statistics.
  • A. Sampling Error: For large sample sizes, the sampling error converges to a zero-mean normal distribution, so its variance characterizes the sampling uncertainty.
  • A. Sampling Error: Sampling uncertainty is estimated for correlated data using an autocorrelation-based variance estimator and effective sample size.The effective sample size is N_eff = N/T_0, where T_0 is the decorrelation separation distance.
  • A. Sampling Error: An autoregressive time-series model estimates the autocorrelation function when direct autocorrelation estimates are noisy, especially for modest sample sizes.Models of increasing order are compared with a finite-sampling information criterion.
  • A. Sampling Error: The AR implementation can experience round-off-related stability problems for large, low-noise datasets or very long decorrelation times.

B. Discretization Error

The discretization-error method extends Richardson extrapolation with a Bayesian model that incorporates sampling uncertainty. This avoids misleading deterministic estimates when statistical sampling error contaminates computed quantities.

  • 2. Accounting for Sampling Error: Bayesian calibration incorporates sampling error into Richardson extrapolation to estimate discretization error probabilistically.
  • 1. Assessing Order of Accuracy without Sampling Error: Classical Richardson extrapolation uses outputs from at least three resolutions assumed to lie in the asymptotic convergence range.The method models the discrete output as the continuum value plus resolution-dependent error terms.
  • 1. Assessing Order of Accuracy without Sampling Error: Standard Richardson extrapolation can produce misleading convergence orders and discretization errors when sampling error is significant.Sampling noise can make the inferred order negative or leave the extrapolation equations unsolvable.
  • 2. Accounting for Sampling Error: The Bayesian model represents the true mean as the sampled mean plus sampling error and a resolution-dependent discretization expansion.
  • 2. Accounting for Sampling Error: Bayesian inference estimates the continuum mean and discretization-model parameters from sample averages computed at distinct mesh sizes.The posterior combines prior information with a likelihood derived from the probabilistic error model.
  • 2. Accounting for Sampling Error: Posterior statistics generally require Markov chain Monte Carlo because the necessary integrals cannot be evaluated analytically.

C. Illustrative Example: The Lorenz Equations

The proposed sampling- and discretization-error estimators are illustrated on means computed from Lorenz-equation solutions. The chaotic system is discretized with fourth-order Runge–Kutta time integration.

  • The estimators are applied to means computed from Lorenz-equation simulations to illustrate their use for statistical quantities.
  • The Lorenz system can exhibit chaotic behavior depending on σ, β, and ρ, making it a simple setting for testing the estimators.
  • The illustrative simulations use typical parameters σ = 10 and β = 8 and discretize the equations with fourth-order Runge–Kutta.

1. Sampling Error Estimator Performance

The correlation-aware sampling-error estimator is consistent with ensemble results for Lorenz averages, while Bayesian Richardson extrapolation distinguishes regimes where sampling uncertainty masks or reveals discretization error.

  • The estimator was consistent with the ensemble standard deviation, with the average estimate differing from the true value by at most 1.2%.Maximum and minimum estimates generally had errors around 10% or less, with a maximum error of 17.5%.
  • Decreasing the sampling interval reduced true uncertainty but eventually increased the estimate below ∆ts ≈ 0.20, consistent with accumulated round-off error.Below ∆ts ≈ 0.01, the estimator algorithm broke down; changing floating-point precision supported the round-off hypothesis.
  • Large sampling uncertainty masked discretization error, leaving the order-of-accuracy posterior essentially unchanged from its prior.The data contained little information about the true discretization error in this regime.
  • With medium uncertainty, sampling error dominated at the smallest timestep, discretization error dominated at the largest, and the data informed the order of accuracy.The posterior peak was near p = 4 but remained substantially uncertain.
  • When sampling error was much smaller than discretization error, Bayesian Richardson extrapolation produced narrow posteriors consistent with standard Richardson extrapolation.The posterior peak for p was approximately 4.24.

III. DNS OF Reτ = 180 CHANNEL FLOW

The channel-flow DNS study applies the error-characterization techniques to Reτ ≈ 180 turbulent flow using nominal, coarser, and finer resolutions across two domain sizes.

  • The simulations analyze fully developed incompressible turbulent channel flow at Reb = 2925 and Reτ ≈ 180.Two domain sizes were used: Lx = 4π, Lz = 2π and Lx = 12π, Lz = 4π.
  • The nominal mesh followed commonly used wall-bounded DNS resolution heuristics, including ∆x+ ≈ 13 and ∆z+ ≈ 7.Two uniform derefinements, labeled coarse and coarsest, were used for Richardson extrapolation.
  • Constant timesteps produced equidistant temporal samples, simplifying the temporal analysis used to estimate sampling uncertainty.The simulations were somewhat better resolved in time than common practice.
  • The small-domain case included a finest grid twice as fine as nominal to validate the Bayesian Richardson extrapolations.The larger domain was selected because its error estimates inform interpretation of a reference Reτ ≈ 180 simulation.

B. Small Domain Results

For centerline mean velocity, the Bayesian model agrees with finest-mesh observations and estimates a very small nominal-mesh discretization error, although sampling uncertainty remains significant.

  • The order-of-accuracy posterior peaked near p = 5 but remained broad, with first and third quartiles near 4.4 and 7.4.
  • The posterior uncertainty in the true centerline mean velocity was less than 0.08% between its 5th and 95th percentiles.This narrow uncertainty persisted despite substantial uncertainty about the order of accuracy.
  • The observed finest-mesh centerline velocity was not near the prediction-distribution tails, so the discretization-error model was not invalidated.The validation compares the observed value with a prediction incorporating calibrated model uncertainty and sampling error.
  • Nearly all nominal-mesh discretization-error probability was below 0.1%, with half below 0.008%.The estimated sampling-error standard deviation was 0.011%, so sampling uncertainty remained significant after more than 2000 flow-throughs.
  • The discretization-error distribution was concentrated on the negative side of zero because the posterior for C was bounded away from zero.Since ǫh = Ch^p, C determines the sign, while weak information about p concentrates probability near zero.

2. Skin Friction

For skin friction and other channel statistics, the model generally predicts small errors, but non-monotonic convergence invalidates it for some quantities and makes discretization error locally comparable to sampling error.

  • 2. Skin Friction: The skin-friction prediction agreed with the finest-mesh observation, and its nominal-mesh discretization error was only 0.3% of the mean.The model was not invalidated for this quantity.
  • 2. Skin Friction: Skin-friction discretization error was large relative to sampling error, whose standard deviation was below 0.05%.
  • Summary of Results for Single-Point Statistics: Streamwise-velocity and vorticity-variance models were invalidated because observed values fell outside 90% credibility intervals over substantial channel regions.Their mesh results were non-monotonic, which the simple model cannot capture; consequently, reliable discretization-error statements were not made.
  • Summary of Results for Single-Point Statistics: For mean velocity and viscous shear stress, both estimated errors were below 1% across the channel, with most median mean-velocity errors below 0.25%.Nearly all 90% credibility intervals for mean velocity were below 0.5%.
  • Summary of Results for Single-Point Statistics: For Reynolds-stress quantities, median discretization error was below 2%, larger near the wall than at the channel center.Near the wall it exceeded sampling error, while near the center the relative ordering was reversed.

C. Large Domain Results

The large-domain simulations use the same domain as Hoyas and Jiménez for direct comparison and uncertainty assessment. Only three meshes were used, preventing validation against a higher-resolution result.

  • The simulations use Lx = 12π and Lz = 4π, matching Hoyas and Jiménez for direct comparison.
  • Only three meshes were used, so the calibrated discretization-error model could not be tested against a higher-resolution result.
  • The large-domain study targets quantities of interest in channel flow and uses assumptions based on comparable Reynolds number and mesh resolution.
  • The simulations collected statistics over only tens of flow-throughs, although each large-box flow-through provided more data than in the small-box case.

1. Centerline Mean Velocity and Skin Friction

For the large-domain channel flow, estimated discretization errors are generally small, but their relative importance varies by quantity and location. Comparisons with prior DNS show discrepancies plausibly attributable to discretization and sampling errors.

  • 0.011% is the mean estimated discretization error for centerline velocity, with a 0.021% median and near certainty that it is below 0.1%.
  • 0.24% is the mean estimated discretization error for skin friction, versus 0.092% for its sampling-error standard deviation.
  • The centerline velocity estimate is 1.16303 ± 0.00024, within 0.031% of Hoyas and Jiménez’s 1.16267 value.
  • The skin-friction estimate is 0.00807834 ± 7.49 × 10^-6, differing by approximately 0.4% from 0.00811666 ± 3.4 × 10^-7, a difference larger than sampling error.
  • Discretization errors are generally below one percent for mean velocity and viscous stress, while errors tend to be largest near the wall and exceed sampling error there.
  • The posterior true-value intervals are small for mean velocity and viscous shear stress, but more visible near peaks of Reynolds stress and velocity variances.

IV. CONCLUSIONS

The paper develops correlated-sampling and Bayesian Richardson tools for systematic DNS uncertainty estimation. Tests support common resolution heuristics for many quantities, while exposing limits for others.

  • The work develops a correlated-sampling error estimator and a Bayesian Richardson-extrapolation extension for discretization error under finite sampling.
  • The tools enable systematic estimation of both sampling and discretization errors in statistical quantities computed from chaotic simulations.
  • The Lorenz-equation results show that the estimators perform well in a simple, well-understood setting.
  • For many channel-flow quantities, including mean velocity and Reynolds shear stress, estimated errors are small and support usual DNS mesh-resolution heuristics.
  • For streamwise velocity variance, model validation fails against higher-resolution simulations, so discretization error cannot be quantified confidently.
  • The conclusions are intended to inform expected numerical accuracy for demanding simulations when combined with sampling-error estimates feasible for expensive DNS.

Appendix A: Motivation for the Sampling Error Estimator

DNS samples are correlated, so classical independent-sample uncertainty estimates are inadequate. The proposed approach extends central-limit reasoning to weakly dependent data by modifying the effective sample count.

  • DNS samples are not independent, especially at small temporal or spatial separations, because their correlation structure is unknown.
  • Downsampling can avoid correlated samples, but choosing the right coarsening is difficult because excessive thinning discards useful data.
  • The estimator seeks uncertainty estimates that extract more information from computationally expensive DNS data.
  • The method extends the central limit theorem from independent identically distributed variables to α-mixing sequences whose distant samples are nearly independent.
  • Weak dependence modifies the effective number of samples through the autocorrelation at separation k.
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