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q-stageRunge - Kutta methoditeration:y y tkn1n i ii1qwithi1k f( y t k , t t)i n ij j n ij1and α 1 = 0explicitimplicit:qk f( y t k , t t)i n ij j n ij199q-stageRunge - Kutta methodq = 1 is the Euler method(unique)Butcher array or Runge-Kuttatableau:2 21 3 31 32 .... q q1 q2 q,q1 .... 1 2 q1qresumes allparametersof the generalRunge-Kutta100
Example RK4Butcher array:its stage is: q = 40½½½ 0 ½1 0 0 11 1 1 16 3 3 6RK4 is of order p = 4 applet101Order of Runge - Kutta methodThe R-K method is of order p if:q( ) p1( ) ii ( )i1y t t y t t k O tOne must choose the α i , β ij and ω i for i,j [1,p] ] suchthat the left hand side = 0 in O(Δt m ) for all m ≤ p.Taylor expansion:1 df y t t y t t O tpm1mp1( ) ( ) ( )m1 m1m! dt y( t),t102
Page 1 and 2: Monte Carlo MethodsNicholas Constan
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Page 9 and 10: M(RT) 2 algorithmN.C. Metropolis, A
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Page 61 and 62: Comparison of methodsΔtfor fixed n

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