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
yn = yn 1 + h biki
; 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"
c1 a11 a12
c2 a21 a22
... ... ... . . . ...
cs as1 as2
or more compactly as cA
b We can also write (3.2.2) in the form
yn = yn 1 + h bi f (tn 1 + cih Yi)
; where Yi = yn 1 + h
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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- Spring '14
- Numerical Analysis, yn, Tn, Numerical ordinary differential equations