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Modeling Basics: 4. Numerical ODE Solving In Excel 5. Solving ODEs in Mathematica By Peter Woolf University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons

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Case study: water storage Water is vitally important for industry, agriculture, domestic consumption. 29.1x10 10 m 3 3.6x10 10 m x10 10 m 3 Data from

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29.1x10 10 m 3 3.6x10 10 m x10 10 m 3 Data from

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29.1x10 10 m 3 3.6x10 10 m x10 10 m 3 Data from Key Question: Have you stored enough water?

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Water storage Water tower P&ID Mathematical Description: Change=input-output

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How to solve this set of equations? Hand calculation Software calculation using Mathematica for example Often not possible for “real” systems Even if possible, sometimes not useful Integrate with software such as Excel or Mathematica (or many others) Integration can be tricky Need to know all parameters Can be computationally demanding Analytical solutionNumerical solution

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Mathematica Syntax DSolve[ {eqn1, eqn2..},{y1,y2,..},t] Analytical solver NDSolve[{eqn1, eqn2..},{y1,y2,..},t] Numerical solver Plot[{f1,f2},{t,0,tmax}] Plotter See lecture7.mathematica.nb

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What if your feed varies in a non-functional way? Time (hours) feed Numerical integration!

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Numerical ODE solving time Time step=0.1 Tank depth (h) True solution 10 Slope: F-k 1 v 1 (10) Euler’s method! Numerical approx

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Numerical ODE solving time Time step=0.1 Tank depth (h) 10 Slope: F-k 1 v 1 (10) Notes on Euler’s method: Perfect fit for linear systems Works well for nonlinear systems if the time step is small (everything is locally linear) If time step is too large can explode Small time steps mean slow calculations

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Numerical ODE solving time Time step=0.1 Tank depth (h) 10 Notes on Euler’s method: Perfect fit for linear systems Works well for nonlinear systems if the time step is small (everything is locally linear) If time step is too large can explode Small time steps mean slow calculations 6.0x Numerical error! Time step=0.01

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Numerical ODE solving time Time step=0.1 Tank depth (h) 10 Is there a way we could use a larger time step but retain accuracy? Solution: take some sort of weighted average of slopes at intermediate points to estimate the next value. Runge-Kutta methods!

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Numerical ODE solving time Time step=0.1 Tank depth (h) 10 4th order Runge-Kutta Where Euler’s method

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Numerical ODE solving time Time step=0.1 Tank depth (h) 10 Why stop at 4th order? 5th order is similar but with different weightings Upsides: 1)Larger step sizes possible 2)Can be more stable 3)Often more computationally efficient Downsides: 1)Complex to implement 2)Similar accuracy can often be achieved using Euler’s method which is easy to implement

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29.1x10 10 m 3 3.6x10 10 m x10 10 m 3 Data from Key Question: Have you stored enough water? See lecture7.excel.xls

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Take home messages Numerical simulation of differential equations can help you model very complex systems Sometimes slower, simpler method are better as your time is often more valuable than a computers Analytical results can be helpful in restricted cases.. Use Mathematica to handle the algebra

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