23b with yi tn 1 cih gives s x ci aij i 1 2 s 325b j

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Unformatted text preview: -order methods. Methods that satisfy (3.2.5b) also treat autonomous and non-autonomous systems in a symmetric manner (Problem 1). We can continue this process to higher orders. Thus, the Runge-Kutta method will be of order p if it is exact when the di erential equation and solution are y = (t ; tn 1)l 0 ; ; y(t) = 1 (t ; tn 1 )l l 1 ; l = 1 2 : : : p: (The use of t ; tn 1 as a variable simpli es the algebraic manipulations.) Substituting these solutions into (3.2.3a) implies that s hl = h X b (c h)l 1 ii l j =1 ; ; or s X bicli 1 = 1 l i=1 l = 1 2 : : : p: ; 11 (3.2.5c) Conditions (3.2.5c) are necessary for a method to be order p, but may not be su cient. Note that there is no dependence on the coe cients aij i j = 1 2 : : : s, in formulas (3.2.5a,c). This is because our strategy of examining simple di erential equations is not matching all possible terms in a Taylor's series expansion of the solution. This, as noted, is a tedious operation. Butcher developed a method of simplifying the work by constru...
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