CS 205A class_16 notes

Scientific Computing

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CS205 - Class 16 Readings : 9.3 Covered in Class: 1, 2, 3, 4, 5, 6 ODE’s (Continued) 1. Recall the model ODE from last time. '( , ) yf t y = a. We stated that 0 λ > is ill-posed. But why? i. Errors accumulated and they increase exponentially. 2. (Forward) Euler’s Method 1 (, ) kk yy f ty h + = or 1 k k y yh f t y + = + a. Accuracy , truncation error usually dominates round-off error in ODE’s. b. Local truncation error 2 1 (, ) ( ) k k h f t y O h + =+ + i. 1 k y + is calculated by ignoring the 2 () Oh term. ii. If 0 y is exact, the error in 1 y is 2 . c. Global truncation error integrating from o tt = to final = with 1/ nO h = steps gives a total error of 2 ( ) Onh = . i. Euler’s method is 1 st order accurate with 1 f ty Oh h + d. For stability consider the model equation ' = where 0 < i. For a general ode is / df dy or an eigenvalue of the Jacobian matrix ii. Euler’s method applied to the model equation is 1 (1 ) k k y y h y + = += + iii. So ) k ko y and the error shrinks when |1 | 1 h + < 1. Thus, 20 h −< < is needed for stability e. Example i. Forward Euler on ' y y =− for 0 1 y = , 0 0 t = . Stability is h <2 ii. h=.5 is stable but truncation errors cause it to be smaller 1.
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CS 205A class_16 notes - CS205 Class 16 Readings 9.3...

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