11 runge kutta methods belong to a class called one

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Unformatted text preview: ssible to write ; 7 them in the general form yn = yn 1 + h (tn 1 yn 1 h): ; ; ; (3.2.1) This representation is too abstract and we'll typically consider an s-stage Runge-Kutta formula for the numerical solution of the IVP (3.1.1) in the form s X yn = yn 1 + h biki (3.2.2a) ; where ki = f (tn 1 + cih yn 1 + h ; i=1 s X ; j =1 aij kj ) i = 1 2 : : : s: (3.2.2b) These formulas are conveniently expressed as a tableau or a \Butcher diagram" a1s c1 a11 a12 c2 a21 a22 a2s ... ... ... . . . ... cs as1 as2 ass b1 b2 bs or more compactly as cA b We can also write (3.2.2) in the form s X yn = yn 1 + h bi f (tn 1 + cih Yi) ; where Yi = yn 1 + h ; s X j =1 ; i=1 aij f (tn 1 + cj h Yj ) ; i = 1 2 : : : s: (3.2.3a) (3.2.3b) In this form, Yi, i = 1 2 : : : s, are approximations of the solution at t = tn + cih that typically do not have as high an order of accuracy as the nal solution yn. An explicit Runge-Kutta formula results when aij = 0 for j i. Historically, all Runge-Kutta formulas were explicit however, implicit formula are very useful for sti systems and problems where solutions os...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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