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To emphasize the di culty well illustrate runge kutta

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Unformatted text preview: : : Ys) = Yi ; yn 1 ; h ; s X j =1 aij f (tn 1 + hcj Yj ) = 0 ; (3.3.9a) and get 2 I ; a11 J(1 ) ;ha12 J(2 ) ;ha1s J(s ) 6 ;ha21 J( ) I ; ha22 J( ) ;ha2s J(s ) 6 1 2 6 ... ... ... ... 4 ;has1 J(1 ) ;has2 J(2 ) I ; hassJ(s ) Yi( +1) = Yi( ) + Yi( ) j = 1 2 ::: s 32 ( ) Y1 ( 7 6 Y2 ) 76 76 . 54 . . i = 1 2 ::: s Ys( ) 3 2 7 7 = ;6 6 7 6 5 4 = 0 1 ::: 3 F(1 ) F(2 ) 7 7 7 ... 5 (3.3.9b) F(s ) (3.3.9c) where J(j ) = fy (tn 1 + hcj Yj( )) ; ( ( F(j ) = Fj (Y1 ) Y2 ) : : : Ys( ) ) j = 1 2 : : : s: (3.3.9d) For an s-stage Runge-Kutta method applied to an m-dimensional system (3.3.7), the Jacobian in (3.3.9b) has dimension sm sm. This will be expensive for high-order methods and high-dimensional ODEs and will only be competitive with, e.g., implicit 25 multistep methods (Chapter 5) under special conditions. Some simpli cations are possible and these can reduce the work. For example, we can approximate all of the Jacobians as J = fy (tn 1 yn 1): ; (3.3.10a) ; In this case, we can even shorten the notation by introducing the Kronecker or...
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