Ordinary Differential Equations (ODEs) 1Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F128 – Zürich

Problem Definition  We are going to look at initial value problems of explicit ODE systems, which means that they can be cast in the following form 2Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Higher Order ODEs  Higher order explicit ODEs can be cast into the first order form by using the following «trick»  Therefore, a system of ODEs of order n can be reduced to first order by adding n-1 variables and equations  Example: 3Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

ODEs as vector field  Since we know the derivatives as function of time and solution, we can calculate them at every point in time and space 4Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

ODE systems as vector fields  We can do the same for ODEs systems with two variables, at a certain time point 5Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Example: A 1-D First Order ODE  Consider the following initial value problem  This ODE and its solution follow a general form (Johann Bernoulli, Basel, 17 th – 18 th century) 6Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Some Nomenclature  On the next slide, the following notation will be used 7Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Numerical Solution of a 1-D First Order ODE  Since we know the first derivative, Taylor expansion suggests itself as a first approach  This is known as the Euler method, named after Leonhard Euler (Basel, 18 th century) 8Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

How does Matlab do it? Non-Stiff Solvers  ode45  ode45 : Most general solver, best to try first; Uses an explicit Runge-Kutta pair (Dormand-Prince, p = 4 and 5)  ode23ode45  ode23 : More efficient than ode45 at crude tolerances, better with moderate stiffness; Uses an explicit Runge- Kutta pair (Bogacki-Shampine, p = 2 and 3).  ode113  ode113 : Better at stringent tolerances and when the ODE function file is very expensive to evaluate; Uses a variable order Adams-Bashforth-Moulton method 9Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

How does Matlab do it? Stiff Solvers  ode15sode45  ode15s : Use this if ode45 fails or is very inefficient and you suspect that the problem is stiff; Uses multi-step numerical differentiation formulas (approximates the derivative in the next point by extrapolation from the previous points)  ode23sode15s  ode23s may be more efficient than ode15s at crude tolerances and for some problems (especially if there are largely different time-scales / dynamics involved); Uses a modified Rosenbrock formula of order 2 (one-step method) 10Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Matlab Syntax Hints [T, Y] = ode45(ode_fun, tspan, y0, …);  All ODE-solvers use the syntax [T, Y] = ode45(ode_fun, tspan, y0, …);  ode_fun is a function handle taking two inputs, a scalar t (current time point) and a vector y (current function values), and returning as output a vector of the derivatives dy/dt in these points  tspan is either a two component vector [tstart, tend] or a vector of time points where the solution is needed [tstart, t1, t2,...]  y0 are the values of y at tstart f_new y, k, p);  Use parametrizing functions to pass more arguments to ode_fun if needed f_new y, k, p);  This can be done directly in the solver call  [T, Y] = y, k, p), tspan, y0); 11Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Exercise 1: Damped harmonic oscillator  A mass on a string is subject to a force when out of equilibrium  According to Newton’s second law (including friction), the resulting movement follows the solution to an ODE: 12Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers m y F Spring force Friction

Assignment 1 1.Rewrite the motion equation for the damped harmonic oscillator as an explicit, first order ODE system ode45 2.Use ode45 to solve the problem function dy = harm_osc(t, y, m, k, a)  Set up a function file to calculate the derivatives, its header should read something like function dy = harm_osc(t, y, m, k, a) m = 1; k = 10; a = 0.5; y0 = 1; dy/dt0 = 0;  Set m = 1; k = 10; a = 0.5; y0 = 1; dy/dt0 = 0; tSpan = [0, 20];  Use tSpan = [0, 20]; 3.Plot the position of the mass y against time 4.Plot the speed of the mass dy/dt against time 13Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Exercise 2: Radioactive decay  A decaying radioactive element changes its concentration according to the ODE where k is the decay constant  The analytical solution to this problem reads 14Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Assignment 2 1.Plot the behavior of the radioactive decay problem as a vector field in the y vs t plane vector_field.m  Find online the function vector_field.m. It plots the derivatives of first order ODEs in different space and time points  Plot the vector field for y values between 0 and 1, time between 0 and 2 and k = 1. Can you see what a solver has to do?  Is it possible to switch from one trajectory in the vector field to another? What follows for the uniqueness of the solutions? 2.Implement the forward Euler method in a function file function [t,y] = eulerForward(f, t0, tend, y0, h)  The file header should read something like function [t,y] = eulerForward(f, t0, tend, y0, h) y0 = 1, k = 1 h = 0.1 t0 = 0 tEnd = 10  Use y0 = 1, k = 1 and h = 0.1 to solve the radioactive decay problem from t0 = 0 to tEnd = 10 and plot the solution together with the analytical solution. h  What happens if you increase the step size h ? What happens if you increase the step size above 2? 15Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Exercise 3: Lotka-Volterra equations  Predator-prey dynamics can be modeled with a simple ODE system: where x is the number of prey, y is the number of predators and a, b, c, d are parameters 16Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers

Assignment 3 1.Plot the behavior of the predator-prey system as a vector field in the x vs y plane vector_field2D.m  Find online the function vector_field2D.m which plots this vector field  Plot x and y between 0 and 1, and use a = c = 0.5 and b = d = 1  Without solving the ODEs, what can you say about the solution behavior just by looking at the trajectories? ode45x0 = y0 = 0.3 tSpan = [0, 50]; 2.Solve the system using ode45, and x0 = y0 = 0.3 as initial values, and tSpan = [0, 50];  Plot x and y against t. Can you observe what you predicted in 1? 17Daniel Baur / Numerical Methods for Chemical Engineers / Explicit ODE Solvers