1 as jzj 1 and these methods are a stable table 331 3

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Unformatted text preview: more generally, methods corresponding to subdiagonal Pade approximations in the rst two bands are L-stable ( 17], Section IV.4). L-stable methods are preferred for sti problems where Re( ) 0 but methods where jR(z)j ! 1 are more suitable when Re( ) 0 but jIm( )j 1, i.e., when solutions oscillate rapidly. Explicit Runge-Kutta methods are easily solved, but implicit methods will require an iterative solution. Since implicit methods will generally be used for sti systems, 24 Newton's method will be preferred to functional iteration. To emphasize the di culty, we'll illustrate Runge-Kutta methods of the form (3.2.3) for vector IVPs y = f (t y ) y(0) = y0 0 (3.3.7) where y, etc. are m-vectors. The application of (3.2.3) to vector systems just requires the use of vector arithmetic thus, s X Yi = yn 1 + h aij f (tn 1 + cj h Yj ) i = 1 2 ::: s (3.3.8a) ; ; j =1 yn = yn 1 + h ; s X i=1 bi f (tn 1 + cih Yi): (3.3.8b) ; Once again, yn etc. are m-vectors. To use Newton's method, we write the nonlinear system (3.3.8a) in the form Fi (Y1 Y2 :...
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