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Unformatted text preview: AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 5: Numerical Solution of Ordinary Differential Equations Dianne P. OLeary c 2002,2004,2006 The Plan Initial value problems (IVPs) for ordinary differential equations (ODEs) Review 460 ODE notes Hamiltonian systems DifferentialAlgebraic Equations Some basics Some numerical methods Boundary value problems for ODEs. Some basics Shooting methods Finite difference methods References: The 460 notes are based on Chapter 9 of Van Loans book. These notes are based on Parts III and IV of Ascher and Petzolds book. Initial value problems for ordinary differential equations Review 460 ODE notes Hamiltonian systems Review 460 ODE notes Hamiltonian systems 1 In some ODE systems, there is an associated conservation principle , and if possible, we formulate the problem so that conservation is observed. Definition: A Hamiltonian system is one for which there exists a scalar Hamiltonian function H ( y ) so that y = D y H ( y ) , where D is a blockdiagonal matrix with blocks equal to J = 1 1 . Example: Linear harmonic oscillator. Let q ( t ) and p ( t ) be unknown functions satisfying q = p p = q where > is a fixed parameter. The Hamiltonian of the system is defined to be H = 2 ( p 2 + q 2 ) . To verify this, note that if y = [ q,p ] T , then y H ( y ) = q p so that y = p q = D y H ( y ) = 1 1 q p . (See http://scienceworld.wolfram.com/physics/HamiltonsEquations.html for more information on Hamiltonian systems.) Note that H = 2 (2 pp + 2 qq ) = 2 (2 q p + 2 p q ) = , so H ( t ) must be constant; in other words, the quantity H is conserved or invariant . We can verify this a different way by writing the general solution to the problem: q ( t ) p ( t ) = cos t sin t sin t cos t q (0) p (0) 2 and computing p ( t ) 2 + q ( t ) 2 . The eigenvalues of the matrix defining the solution are imaginary numbers, so a small perturbation of the matrix can cause the quantity H to either grow or shrink, and this will not produce a useful solution....
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This note was uploaded on 02/05/2008 for the course CMSC 660 taught by Professor Oleary during the Fall '06 term at Maryland.
 Fall '06
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