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Lattice gasParticles move on a triangular lattice andfollow the following collision rules:Momentum is conserved at each collision.It can be proven (Chapman-Enskog) thatits continuum limit is the Navier Stokes eq.71von Karman streetvelocity field of a fluid behind an obstacleEach vector is an average over time and space.72
Solving equationsFinding the solution (root)) of an equation:f (x)) = 0is equivalent to the optimization problemof finding the minimum (ormaximum) ) of F(x)given by:d F ( x )dx073Newton methodBe x 0 a first guess, then linearize around x 0 :f( x ) f( x ) ( x x ) f '( x ) 01 0 1 0 0xx n1nf( x )fn'( x )n74
Page 1 and 2: 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: Slow coolingT 0T 1T 20Different coo
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 and 56: Adaptive time-stepΔtδ expectedDef
Page 57 and 58: Richardson extrapolationCalculate f
Page 59 and 60: 3th order predictor- correctoruse e
Page 61 and 62: Comparison of methodsΔtfor fixed n

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