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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 39 Dec 2, 2011 Prof. Kamesh Madduri Class Overview Eulers method Slides from textbook follow. 2 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Eulers Method Accuracy and Stability Implicit Methods and Stiffness Eulers Method For general system of ODEs y = f ( t, y ) , consider Taylor series y ( t + h ) = y ( t ) + h y ( t ) + h 2 2 y 00 ( t ) + = y ( t ) + h f ( t, y ( t )) + h 2 2 y 00 ( t ) + Eulers method results from dropping terms of second and higher order to obtain approximate solution value y k +1 = y k + h k f ( t k , y k ) Eulers method advances solution by extrapolating along straight line whose slope is given by f ( t k , y k ) Eulers method is singlestep method because it depends on information at only one point in time to advance to next point Michael T. Heath Scientific Computing 22 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Eulers Method Accuracy and Stability Implicit Methods and Stiffness Example: Eulers Method Applying Eulers method to ODE y = y with step size h , we advance solution from time t = 0 to time t 1 = t + h y 1 = y + hy = y + hy = (1 + h ) y Value for solution we obtain at t 1 is not exact, y 1 6 = y ( t 1 ) For example, if t = 0 , y = 1 , and h = 0 . 5 , then y 1 = 1 . 5 , whereas exact solution for this initial value is y (0 . 5) = exp(0 . 5) 1 . 649 Thus, y 1 lies on different solution from one we started on Michael T. Heath Scientific Computing 23 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Eulers Method Accuracy and Stability Implicit Methods and Stiffness Example, continued To continue numerical solution process, we take another step from t 1 to t 2 = t 1 + h = 1 . , obtaining y 2 = y 1 + hy 1 = 1 . 5 + (0 . 5)(1 . 5) = 2 . 25 Now y 2 differs not only from true solution of original problem at t = 1 , y (1) = exp(1) 2 . 718 , but it also differs from solution through previous point ( t 1 , y 1 ) , which has approximate value 2 . 473 at t = 1 Thus, we have moved to still another solution for this ODE Michael T. Heath Scientific Computing 24 / 84 Ordinary Differential Equations...
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This note was uploaded on 01/19/2012 for the course CMPSC 451 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.
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