Unformatted text preview: unge-Kutta
and collocation methods. With u(t) a polynomial of degree s in t for t tn 1, a collocation
method for the IVP
; y = f (t y) y(tn 1) = yn 0 ; ; (3.3.12a) 1 consists of solving u(tn 1) = yn
; (3.3.12b) 1 ; u (tn 1 + cih) = f (tn 1 + cih u(tn 1 + cih)) i = 1 2 ::: s 0 ; ; ; (3.3.12c) where ci, i = 1 2 : : : s, are non-negative parameters. Thus, the collocation method
consists of satisfying the ODE exactly at s points. The solution u(tn 1 + h) may be used
as the initial condition yn for the next time step.
Usually, the collocation points tn 1 + ci h are such that ci 2 0 1], i = 1 2 : : : s, but
this need not be the case 6, 10, 20].
Generally, the ci, i = 1 2 : : : s, are distinct and we shall assume that this is the case
here. (The coe cients need not be distinct when the approximation u(t) interpolates
some solution derivatives, e.g., as with Hermite interpolation.) Approximating u (t),
t tn 1, by a Lagrange interpolating polynomial of degree s ; 1, we have
u (t) = kj Lj ( t ;hn 1 )
; ; 0 ; ; 0 w...
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- Spring '14