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Ch E 310 - Fall 10 - Lecture 24

# Ch E 310 - Fall 10 - Lecture 24 - Lecture 24 Agenda Initial...

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Lecture 24 – November 30, 2010 Agenda: Initial Value Problems (solving ODEs) ODE “Stiffness” Explicit vs. Implicit Methods Systems of stiff ODEs MATLAB functions for handling stiff ODEs

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Stiffness A “stiff” system is one that has both rapidly and slowly varying components In some cases, the rapid components decay over time and solution is eventually dominated by the slower components Even if the above is true, the solution will necessitate a step size that can accommodate both components An example of a stiff ODE: dy dt   1000 y 3000 2000 e t
Stiffness where y (0) = 0 : How do we solve this kind of ODE analytically? Recall the general form for linear first order ODEs: First, calculate the integrating factor m ( t ) = exp[ p ( t ) dt ] Next, multiply both sides by m ( t ) and integrate Last, solve for y ( t ) and apply the initial condition     t q y t p y dy dt   1000 y 3000 2000 e t

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Stiffness Solving the ODE where y (0) = 0 gives: A plot of the analytical solution shows the problem: The solution has fast and slow components, as captured by the exponential decay constants (-1000 and -1) y 3 0.998 e 1000 t 2.002 e t
Recall the stability criterion established for Euler’s Method: h < 2/ a (for dy / dt = –ay ) The larger a is, the smaller the step size must be In the present example a ~ 1000 h ~ 0.002 As the fast transient dies out, the step size can increase, however if h is fixed, the solution will take longer We have another remedy for problems of this sort: implicit methods “Implicit” means that the unknown appears on both sides of the equation (like many of the non-linear functions we’ve previously solved using root finding) Stiffness: Implicit Methods

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An implicit Euler Method is derived as follows:
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Ch E 310 - Fall 10 - Lecture 24 - Lecture 24 Agenda Initial...

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