A stochastic (Langevin) simulation of a sedimenting spherical particle reports a…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Scientific Computing for Physicists, Ch The uncertainty of a simulated number: inputs, sensitivity and correlated output

Problem

A stochastic (Langevin) simulation of a sedimenting spherical particle reports a terminal velocity time series of

N = 50000

samples, sample mean

\bar v_{\rm sim} = 1.243\,\mathrm{mm\,s^{-1}}

and sample standard deviation

\sigma = 0.180\,\mathrm{mm\,s^{-1}}

. Fitting an exponential to the series' autocorrelation function (computed by the fast Fourier transform, as in the spectral-estimation chapter) gives an AR(1)-like decay with

\phi = 0.75

. The simulation's other uncertainty contributions have already been assembled: a numerical uncertainty

u_{\rm num} = 0.0050\,\mathrm{mm\,s^{-1}}

from a grid-convergence study (Richardson extrapolation from two grids, assuming the formal order

p=2

), and an input uncertainty

u_{\rm input} = 0.0100\,\mathrm{mm\,s^{-1}}

propagated linearly from the fluid viscosity and particle radius. An independent experiment gives the terminal velocity as

D = 1.280\,\mathrm{mm\,s^{-1}}

with uncertainty

u_D = 0.0200\,\mathrm{mm\,s^{-1}}

. ((a)) Using

\tau_{\rm int} = \tfrac12 + \phi/(1-\phi)

and

s = (1+\phi)/(1-\phi)

, find

\tau_{\rm int}

and

s

for this series. ((b)) Find the statistical uncertainty

u_{\rm stat}

of the simulated mean velocity. ((c)) Assemble the combined simulation uncertainty

u_{\rm sim}

from

u_{\rm num}

u_{\rm stat}

and

u_{\rm input}

. ((d)) Following the comparison-error method (validation), compute

E = \bar v_{\rm sim} - D

and the validation uncertainty

u_{\rm val} = \sqrt{u_{\rm sim}^2 + u_D^2}

. With a coverage factor

k=2

, state whether the comparison can tell the simulation from the experiment.

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