Ch E 310 - Fall 10 - Lecture 23

Ch E 310 - Fall 10 - Lecture 23 - Lecture 23 November 16,...

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Unformatted text preview: Lecture 23 November 16, 2010 Agenda: Initial Value Problems (IVPs) solving ODEs Specialized Runge-Kutta Methods o Adaptive Runge-Kutta o MATLAB: ode23 and ode45 o Solving coupled ODEs with ode45 Analytical versus numerical solutions Self-Check: Solving a simple IVP Adaptive Methods To this point, we have considered solving ODEs with a constant step size h This can be a problem if we encounter a system like this: A smaller step size h is needed for abrupt changes MATLAB function ode23 : uses the BS23 algorithm, simultaneously using second and third order RK formulas to solve the ODE and make error estimates for the step size adjustment The formulas have a familiar (higher order) RK form: MATLAB ODE solving: ode23 h k y h t f k h k y h t f k y t f k h k k k y y i i i i i i i i 2 3 1 2 1 3 2 1 1 4 3 , 4 3 2 1 , 2 1 , 4 3 2 9 1 The error is estimated as: After each step, the error is checked to see if it is within a given tolerance If it is, y i +1 is accepted and k 4 becomes k 1 in the next step If not, the step is repeated with reduced step sizes until the error satisfies: where RelTol is relative tolerance (default 10 3 ) and AbsTol is absolute tolerance (default 10 6 ) 1 1 4 4 3 2 1 1 , 9 8 6 5 72 1 i i i...
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Ch E 310 - Fall 10 - Lecture 23 - Lecture 23 November 16,...

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