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ODE notes - AMSC/CMSC 660 Scientific Computing I Fall 2006...

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AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 5: Numerical Solution of Ordinary Differential Equations Dianne P. O’Leary c 2002,2004,2006 The Plan Initial value problems (IVPs) for ordinary differential equations (ODEs) Review 460 ODE notes Hamiltonian systems Differential-Algebraic 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 Loan’s book. These notes are based on Parts III and IV of Ascher and Petzold’s book. Initial value problems for ordinary differential equations Review 460 ODE notes Hamiltonian systems Review 460 ODE notes Hamiltonian systems 1
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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 0 = D y H ( y ) , where D is a block-diagonal matrix with blocks equal to J = 0 1 - 1 0 . Example: Linear harmonic oscillator. Let q ( t ) and p ( t ) be unknown functions satisfying q 0 = ωp p 0 = - ωq where ω > 0 is a fixed parameter. The Hamiltonian of the system is defined to be H = ω 2 ( p 2 + q 2 ) . To verify this, note that if y = [ q, p ] T , then y H ( y ) = ωq ωp so that y 0 = ωp - ωq = D y H ( y ) = 0 1 - 1 0 ‚ • ωq ωp . (See http://scienceworld.wolfram.com/physics/HamiltonsEquations.html for more information on Hamiltonian systems.) Note that H 0 = ω 2 (2 pp 0 + 2 qq 0 ) = ω 2 (2 q 0 ω p 0 + 2 - p 0 ω q 0 ) = 0 , 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 ωt sin ωt - sin ωt cos ωt ‚ • q (0) p (0) 2
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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. [] Therefore, in solving systems involving Hamiltonians (conserved quantities), it is important to build conservation into the numerical method whenever possible!
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