A numerical check of pattern amplitude saturation is planned for the Swift-Hohenberg…
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength
Problem
A numerical check of pattern amplitude saturation is planned for the Swift-Hohenberg equation ∂tu=ru−(1+∂x2)2u−u3 at r=0.2, run from small random initial data, one independent realisation per random seed.
((a)) Find the unstable band of wavenumbers.
((b)) Find the predicted saturated amplitude A∗.
((c)) A single realisation's measured peak amplitude has standard deviation 0.03 about the true A∗, due to finite-size and finite-time effects. Using Chapter 1's Monte Carlo result that the standard error of a mean over n independent realisations scales as σ/n, find the minimum number of independent realisations needed to pin down the mean amplitude to a standard error of 0.005.