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Error estimationimprove systematically:2yt t y O t2 2example RK4:p2( ) 2 ( )p1ytt y Ot15p2( )2( )111Error estimationGeneral for two different time steps Δt 1 and Δt 2pp1y ytti O tii , 1,2ip p p pp1t 2t1 y t2 y p t1y p O titt1 2t y t yy O tpp2 p 1 pt 1t2p1p pit2 t1112
Richardson extrapolationCalculate forvarious Δt iand extrapolatey =limytt i 0i113Bulirsch - Stoer MethodExtrapolate with rational function:ykp0 p1tn ... pk tnt nm tn0q0 q1tn ...qm tnR. Bulirsch and J. Stoer, „Introduction toNumerical Analysis“ (Springer, NY, 1992)ttn , n2, 4, 6,8,12,16, 24,32, 48, 64,96,. ..nChoose k and m appropiately.y114
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Monte Carlo MethodsNicholas Constan
Page 3 and 4:
Calculation of π1x 2 + y 2 = 1 124
Page 5 and 6: Error of integrationconventional me
Page 7 and 8: Not smooth integralsg(x)ax ineed mo
Page 9 and 10: M(RT) 2 algorithmN.C. Metropolis, A
Page 11 and 12: Detailed balancedp( X , t)dt pY ( )
Page 13 and 14: The Ising ModelBinary variables:i1,
Page 15 and 16: Binary mixtures(lattice gas)Conside
Page 17 and 18: Ising interface+++ +NH E J ij+i, j
Page 19 and 20: Ising interfaceAdd next-nearest nea
Page 21 and 22: Irreversible growth• T. Vicsek,
Page 23 and 24: Random deposition with surface diff
Page 25 and 26: Ballistic depositionParticles fall
Page 27 and 28: DLA clustersfractaldimensiond f = 1
Page 29 and 30: Dielectric breakdown57Dielectric br
Page 31 and 32: Dielectric breakdown modelη = 0.16
Page 33 and 34: Simulated Annealing (SA)S. Kirckpat
Page 35 and 36: Slow coolingT 0T 1T 20Different coo
Page 37 and 38: Solving equationsFinding the soluti
Page 39 and 40: Bisection methodTake two starting v
Page 41 and 42: N-dimensionalequationsBex a N-dimen
Page 43 and 44: N-dimensionalsecant methodIf the de
Page 45 and 46: Euler methodStart with y 0 and iter
Page 47 and 48: The order of a methodIf the error a
Page 49 and 50: Fourth order Runge - Kuttak1tf( y ,
Page 51 and 52: Example RK4Butcher array:its stage
Page 53 and 54: Example q = p = 2y ny t ( 1 )
Page 55: Adaptive time-stepΔtδ expectedDef
Page 59 and 60: 3th order predictor- correctoruse e
Page 61 and 62: Comparison of methodsΔtfor fixed n
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