An explicit finite-difference diffusion code with a random source term at every grid point

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

An explicit finite-difference diffusion code with a random source term at every grid point, drawn afresh each step from a seeded generator, is run until the density field is statistically steady. It then records the local density at one grid point once per step,

N = 8192

samples, with sample variance

\sigma^2 = 0.85

(in units of a reference density, squared). The autocorrelation function of the series, computed via the fast Fourier transform (the Wiener-Khinchin route of the spectral-estimation chapter), is

k

: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

\hat\rho(k)

: 0.62, 0.39, 0.24, 0.15, 0.09, 0.06, 0.04, 0.02, 0.01, 0.01,

-0.01

, 0.00, 0.01,

-0.01

Separately, the mean local density itself was recomputed on grids with

\Delta x

and

\Delta x/2

(diffusion number

\alpha \le \tfrac12

throughout); Richardson extrapolation, assuming the scheme's formal order

p = 2

, gives it a numerical uncertainty

u_{\rm num} = 0.0150

in the same density units. ((a)) Apply the chapter's self-consistent window rule, the smallest

M

with

M \ge 6\,\hat\tau_{\rm int}(M)

where

\hat\tau_{\rm int}(M) = \tfrac12 + \sum_{k=1}^{M}\hat\rho(k)

, to find

\hat\tau_{\rm int}

and the statistical inefficiency

s = 2\hat\tau_{\rm int}

. Say what stopping the sum at lag

6

would have given. ((b)) Find the effective sample size and the statistical uncertainty

u_{\rm stat}

of the mean local density. ((c)) Combine

u_{\rm num}

and

u_{\rm stat}

into the mean density's combined uncertainty

u_{\rm sim}

Browse the problem library