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Example: Lorenz equationIs a simplified system of equations describing the 2-dimensional2flow of fluid of uniform depth in the presence of an imposedtemperature difference taking into account gravity, buoyancy,thermal diffusivity, and kinematic viscosity (friction).σ Prandtl number Rayleigh numberσ = 10 , = 8/3 is varied.Chaos for = 28.'y =σ y - y1 2 1'y = y ρ -y -y2 1 3 2'y = y y - βy3 1 2 3Edward Norton Lorenz(1963)appletC.Sparrow: „The Lorenz Equations: Bifurcations, Chaosand Strange Attractors“ (Springer Verlag, N.Y., 1982)107Example: Lorenz equationChaotic solutions of theLorenz equation exist andare not numerical artefacts(14th math problem of Smale).Warwick Tucker(2002)108
Adaptive time-stepΔtδ expectedDefine the errorThen measure the realdefine a newtime-stepthrough:tyou want to accept.error andreal errorbecause δ Δt p+1.δ measuredexpectednewtold measured1p1109Error estimationLet us consider a method of order p, (error = ΦΔt p+1) .Be y 1 the predicted value for 2Δtand y 2 the predicted value making two steps Δt.Define error as δ = y 1 – y 2y1 2t O tyt ( 2t) y22t O t 2 2t O tp1 p2( ) ( )p1 p2( ) ( )p1 p1 p2( ) ( )110
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Monte Carlo MethodsNicholas Constan
Page 3 and 4: Calculation of π1x 2 + y 2 = 1 124
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Page 7 and 8: Not smooth integralsg(x)ax ineed mo
Page 9 and 10: M(RT) 2 algorithmN.C. Metropolis, A
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