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class_16 - CS205 - Class 16 Readings: 9.3 Covered in Class:...

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CS205 - Class 16 Readings : 9.3 Covered in Class: 1, 2, 3, 4, 5, 6 ODE’s (Continued) 1. Model ODE Problems a. Scalar ODE ' ( , ) y f t y and the i. linear model ODE is ' y y which solution is 0 ( ) 0 t t y y e ii. Only three kinds of solutions iii. 0 ill-posed unstable 1 2 3 4 5 2.5 5 7.5 10 12.5 15 17.5 20 iv. 0 stable, well-posed 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 v. 0 linearly stable 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 b. Vector ODE ' ( , ) y f t y    and the linear model ODE is y J y   . Here is where it gets more interesting as the characterization of the ODE is dependent on the eigenvalues of the Jacobian matrix. 2. Recall the model ODE from last time. ' ( , ) y f t y a. We stated that 0 is ill-posed. But why? i. Errors accumulated and they increase exponentially. 3. (Forward) Euler’s Method 1 ( , ) k k k k y y f t y h or 1 ( , ) k k k k y y hf t y a. Accuracy , truncation error usually dominates round-off error in ODE’s. b.
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class_16 - CS205 - Class 16 Readings: 9.3 Covered in Class:...

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