Non autonomous odes can be written as autonomous

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Unformatted text preview: unge-Kutta methods of order p = 1 2 3 4 (interiors of smaller closed curves to larger ones). 17 1 by letting t be a dependent variable satisfying the ODE t = 1. A Runge-Kutta method for an autonomous ODE can be obtained from, e.g., (3.2.3) by dropping the time terms, i.e., s X yn = yn 1 + h bif (Yi) 0 ; with Yi = yn 1 + h ; s X j =1 i=1 aij f (Yj ) i = 1 2 : : : s: The Runge-Kutta evaluation points ci , i = 1 2 : : : s, do not appear in this form. Show that Runge-Kutta formulas (3.2.3) and the one above will handle autonomous and non-autonomous systems in the same manner when (3.2.5b) is satis ed. 3.3 Implicit Runge-Kutta Methods We'll begin this section with a negative result that will motivate the need for implicit methods. Lemma 3.3.1. No explicit Runge-Kutta method can have an unbounded region of abso- lute stability. Proof. Using (3.2.10), the region of absolute stability of an explicit Runge-Kutta method satis es jyn=yn 1j = jR(z)j 1 z=h ; where R(z) is a polynomial of degree s, the number of stages of the method. Since R(z) is a polynomial, jR(z)j ! 1 as jzj ! 1 and, thus, the stability region...
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