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
Case study: water storage Water is vitally important for industry, agriculture, domestic consumption. 12.1x10 10 m 3 3.6x10 10 m 3 29.1x10 10 m 3 Data from http://www.newton.dep.anl.gov/askasci/gen01/gen01629.htm
12.1x10 10 m 3 3.6x10 10 m 3 29.1x10 10 m 3 Data from http://www.newton.dep.anl.gov/askasci/gen01/gen01629.htm
Key Question: Have you stored enough water? 12.1x10 10 m 3 3.6x10 10 m 3 29.1x10 10 m 3 Data from http://www.newton.dep.anl.gov/askasci/gen01/gen01629.htm
Water storage Water tower P&ID Mathematical Description: Change=input-output dh dt = F " k 1 v 1 h h[0] =10
How to solve this set of equations? dh dt = F " k 1 v 1 h h[0] =10 Numerical solution 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 solution Hand calculation Software calculation using Mathematica for example Often not possible for real systems Even if possible, sometimes not useful
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
Time (hours) 0 0.5 1 1.5 2 2.5 3 3.5 What if your feed varies in a non-functional way? feed 1.0 1.1 1.3 1.4 1.2 0.9 Numerical integration! 0.8 1.1
Numerical ODE solving dh dt = F " k 1v 1 h h[0] =10 True solution Numerical approx Tank depth (h) Slope: F-k 1 v 1 (10) dh dt " #h #t = h $ h i+1 i #t = F $ k 1 v 1 h h i+1 = h i + "t F # k 1 v 1 h [ ] 10 0 0.1 time Time step=0.1 Euler s method!
Numerical ODE Tank depth (h) solving 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 10 0 0.1 time Time step=0.1
Numerical 6.0x10 3023 ODE Tank depth (h) Numerical error! solving 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 10 0 0.1 time Time step=0.1 Time step=0.01
Numerical ODE Tank depth (h) solving Runge-Kutta methods! Is there a way we could use a larger time step but retain accuracy? h i+1 = h i + "t slopes [ ] Solution: take some sort of weighted average of slopes at intermediate points to estimate the next value. 10 0 0.1 time Time step=0.1
Numerical ODE solving h i+1 = h i + "t slopes [ ] 4th order Runge-Kutta slopes = 1 [ 6 k 1 + 2k 2 + 2k 3 + k 4 ] Tank depth (h) 10 0 0.1 time Time step=0.1 Where k 1 = f (0,h o ) Euler s method k 2 = f (0 + "t /2,h o + k 1 "t /2) k 3 = f (0 + "t /2,h o + k 2 "t /2) k 4 = f (0 + "t,h o + k 3 "t)
Numerical ODE Tank depth (h) 10 solving 0 0.1 time Time step=0.1 h i+1 = h i + "t slopes [ ] 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
Key Question: Have you stored enough water? 12.1x10 10 m 3 See lecture7.excel.xls 3.6x10 10 m 3 29.1x10 10 m 3 Data from http://www.newton.dep.anl.gov/askasci/gen01/gen01629.htm
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