1. Ordinary Differential Equations (ODEs) 1Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F128 – Zürich E-Mail: daniel.baur@chem.ethz.ch http://www.morbidelli-group.ethz.ch/education/index

2. Some repetition  Taylor expansion (linearization):  In multiple dimensions:  Inequalities with absolute values  Convergence of linear loops converges if  Spectral radius of compound matrices 2Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

3. Stiffness of non-linear problems  There exist some methods to predict stiffness of a problem, but only for linear problems, e.g. large stiffness ratio, or different time scales of solution behaviors: 3Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

4. Stiffness of non-linear problems  If we have a non-linear problem, one approach is to linearize it and look at the resulting linear problem 4Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

5. Solver tolerances  As mentioned last time, sophisticated solvers adapt their step size (and sometimes their order) to fulfill certain criteria on the error estimation  In Matlab, the error is estimated at every time point and it has to fulfill  |e(i)| ≤ max(RelTol*abs(y(i)),AbsTol(i))  The defaults ( RelTol = 1e-3, AbsTol = 1e-6 ) can be adjusted with the commands  options = odeset('abstol',1e-8,'reltol',1e-9); [T,Y] = ode45(@odefun,tSpan,y0,options); 5Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

6. Assignment 1 1.Find the stability region for the second order Heun method (by hand)  The second order Heun method reads  Use a linear, constant coefficient ODE as test system, i.e.  Find an expression of the form abs(...) < 1 that guarantees stability if it is fulfilled 6Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

7. Assignment 1 (continued) 2.Try to use the definition of the complex absolute value to find an expression that does not contain the absolute value operators and use the function stability_region (given online) to plot the stability region.  The easiest approach is to substitute a complex number w for h* λ max, defined as w = a + b*i, and then applying the definition of the complex absolute value:  Note that for this, you will have to multiply out the square of w (remember that i 2 = -1). Before you resolve the absolute value, bring the complex number into standard form c + d*i 7Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

8. Exercise 2  Consider the following chemical reactions in a batch reactor: 8Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

9. Assignment 2 1.Solve the batch reactor and plot the solutions  Use k1 = 1000; k2 = 0.1; y0 = [1, 0.1] and tSpan = [0,1];  Do you expect the system to be stiff? Use the linearization method to check the stiffness ratio at y = y0 (by hand), by calculating the eigenvalues of the Jacobian of the system  Use ode45 to solve the system and plot the solutions. Is it stiff? Looking at the Jacobian, can you say why? Note how y changes and how it changes the Jacobian and its eigenvalues! 9Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

10. Exercise 3  The van der Waals equation for some non-ideal gas reads  Taking the derivative, we get 10Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers

11. Assignment 3 1.Plot the van der Waals equation for 1000 points between V = 0.34 and V = 4 2.Try to approximate the equation by solving the ODE  Use ode45, tSpan = [0.34, 4]; and y0 = P(0.34) ;  Note that the ODE does not depend on the solution, but only the independent variable (i.e. the first input into your odefun!)  Plot the solution of the ODE together with the analytical solution and zoom in to ylim([0, 2]). What do you observe?  Try the same with ode15. What do you observe when you zoom in? 3.Plot dP/dV against V in the range we considered.  What might be the problem with the solvers, considering what they have to do in the slope field? 4.Tighten both tolerances to 1e-9.  Plot the solutions again and zoom in to ylim([0,2]). 11Daniel Baur / Numerical Methods for Chemical Engineers / Implicit ODE Solvers