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5th order predictor- corrector1st order eq.:drdt f(r) r f(r ) r r rC P C P1 0 1 12nd order eq.:Corrector:d rdt2C P C P f(r) r 2 22f(r 0) r r2 r2119Gear coefficients (1971)1st order equation:2nd order equation:120
Comparison of methodsΔtfor fixed number of iterations nVerlet4th5th6therrory n+1- y nPredictor - Corrector121Sets of coupled ODEsExplicit Runge Kutta and Predictor Correctormethods can be straightforwardly generalizedto a set of coupled 1st order ODEs:dydtif ( y ,..., y , t) , i ,..., Ni1 N1by inserting simultaneously all the valuesof the previous iteration.122
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Monte Carlo MethodsNicholas Constan
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Calculation of π1x 2 + y 2 = 1 124
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Error of integrationconventional me
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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
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Page 19 and 20: Ising interfaceAdd next-nearest nea
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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
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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 )
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Page 57 and 58: Richardson extrapolationCalculate f
Page 59: 3th order predictor- correctoruse e
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