{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 11 runge kutta methods belong to a class called one

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online