1. Quadrature rules 1Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature Michael Sokolov ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F123 – Zürich E-Mail: michael.sokolov@chem.ethz.ch http://www.morbidelli-group.ethz.ch/education/index

2. 2Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature Quadrature methods Single Step Trapezoidal Rule Composite Trapezoidal Rule

3. Composite Midpoint Rule 3Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature ax1x1 x2x2 x n-1 b Constant function for each step

4. Composite Trapezoidal rule 4Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature ax1x1 x2x2 x n-1 b Linear function for each step

5. Composite Simpson rule 5Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature The interval is split up and the areas are integrals of quadratic functions ax1x1 x2x2 x n-1 b Parabola through f(a), f(x1), f(x2)

6. Gauss Quadrature 6Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature Depending on the polynomial order n nodes x j and weights w j are used To approximate the area under a function. n = 3

7. Degree of exactness  Trapezoids are areas under linear functions  Linear functions are approximated exactly; q = 1  Simpson uses the area under quadratic functions  Polynomials up to order three are approximated exactly! q = 3  Even degree interpolation polynomials get one degree of exactness for free  Example 7Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

8. Degree of exactness vs. order of accuracy  When a non-exact result is obtained, the error is proportional to the step size to a certain power s, the order of accuracy  It can be shown that s = q + 1 for sufficiently smooth f 8Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

9. Solution of Nonlinear Functions 9Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature Michael Sokolov ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F123 – Zürich E-Mail: michael.sokolov@chem.ethz.ch http://www.morbidelli-group.ethz.ch/education/index

10. Zero of a Nonlinear Function  Problem definition:  Find the solution of the equation f(x) = 0 for scalar valued f and x; Look for the solution either in  An interval, generally – ∞ < x < ∞  In the uncertainty interval [a, b], where f(a)f(b) < 0  Types of algorithms available: 1.Bisection method 2.Substitution methods 3.Methods based on function approximation  Assumptions:  In the defined intervals, at least one solution exists  We are looking for one solution, not all of them 10Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

11. Bisection Method 1.Define starting interval [a,b] (check that f(a)*f(b) < 0) 2.Compute x = mean([a, b]) 3.Redefine the interval  Set either a = x or b = x so that f(a)*f(b) < 0 is still fulfilled 4.Iterate 2 and 3 until the requested precision is reached  Advantages  After n iterations, the interval is reduced by 2 n  Final precision can be predicted a priori  Disadvantages  Function characteristics are not used to speed up the algorithm 11Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature a b x a

12. Newton Method  The Newton method is based on Taylor expansion  Advantages  Theoretically fastest convergence  Disadvantages  Convergence is not guaranteed even if the uncertainty interval is known  If the derivative must be calculated numerically, the secant method is more convenient 12Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature x0x0 x1x1

13. Secant Method  The Secant is based on the same principles as the Newton method, but it approximates the derivative numerically  Advantages  Does not require the analytical first order derivative  Disadvantages  Convergence is not assured even if the uncertainty interval is known  Convergence is slower 13Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature x 0 = 1.8 x 1 = 1.7 SecantNewton

14. How does Matlab do it? Nonlinear Functions  fzero  fzero finds the zero of a scalar valued function; It uses a combination of bisection, secant, and inverse quadratic interpolation methods  roots  roots finds all the roots of a polynomial function; It computes the eigenvalues of the companion matrix, which correspond to the roots of the polynomial  A = diag(ones(n-1,1),-1); A(1,:) = -c(2:n+1)./c(1); eig(A);  Where c is the vector of the polynomial coefficients 14Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

15. Matlab Syntax Hints  x = fzero(fun, x0);  fun is a function taking as input a scalar x, returning as output the scalar function value f(x)  x0 is either an initial guess (if it has length 1) or an uncertainty interval (if it has length 2, then f(x0(1))*f(x0(2)) < 0 must be fulfilled)  x = roots(c);  c is a vector containing the polynomial coefficients in the order 15Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

16. Assignment 1: Quadrature method comparison  Consider the function and its integral in the range [-2,2]  Using a discretization x k = a + hk with h = (b-a)/(N-1) and k = 0,1,…,N, the quadratures are given by  The Gaussian quadrature is defined for the polynomial order n as: 16Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature Midpoint Trapezoidal Simpson (1)(2) (3) (4) (5) (6)

17. Assignment 1: Quadrature method comparison gauss_int(f,N,a0,b0) 1.Define a new function of the form gauss_int(f,N,a0,b0) where you discretize x and evaluate the overall area as a sum of subareas evaluated with the Gaussian quadrature above. Note that for n = 3 and. logspace(1, 5, 100) 2.In your main file, vary N between 10 and 10 5 using logspace(1, 5, 100), and calculate the relative absolute error of the four approximations compared to the analytical solution of (2) for each h. loglog 3.Plot h vs. the relative errors using loglog for the four methods. polyfit 4.The order of accuracy can be determined as the slope of the double- logarithmic plot. Use polyfit to obtain the corresponding slope for each of the methods. a.In the case of non-linear behavior reduce the fitting to the linear area. Why can the non-linear behavior at very small relative errors be neglected? b.Compare your results with the rules from the lecture. fprintf 5.Provide the output in an appropriate format using fprintf. 6.Think of an alternative way to implement the Gaussian quadrature for a given number of approximation points N. 17Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

18. Exercise: CSTR Multiple Steady States  Consider a CSTR where a reaction takes place  We assume the following  V = const., i.e. Q in = Q out = const.  Perfect coolant behavior, i.e. T C,in = T C,out = const.  Constant density and heat capacity of the reaction mixture  Constant reaction enthalpy  Constant feed, i.e. c A,in = const., T in = const. 18Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

19. CSTR Mass and Energy Balances  The mass and energy balances read  With the T-dependent reaction rate constant 19Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

20. Dimensionless Mass and Energy Balances  If we define  We get a dimensionless form 20Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

21. CSTR Temperature Equilibrium  The steady state concentration of A reads  The temperature in steady state is therefore given by 21Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

22. Assignment 2 1.Plot the total heat flow from and to the reactor vs. the dimensionless reactor temperature  Use α = 49.46; κ 0 = 2.17e20; K C = 0.83; η = 0.33 and θ C = 0.9. 2.Implement and use the secant method to find the three steady state temperature of the CSTR. function [x,xvec] = secantRoot(f,x0)  Use a function of the form function [x,xvec] = secantRoot(f,x0) xvec  Also return the x-values calculated as a vector xvec.  The calculation steps of the secant method can be found on slide 7  The secant method uses two starting guesses; from x0, calculate x1 = (1+ε)*x0. Suggest a value for ε (not too small). abs(x k – x k-1 ) > 1e-8 f(x k ) > 1e-6 n 1e-8 and f(x k ) > 1e-6 and n < 1e5  You will have to store two x-values at any given iteration, that is x k and x k-1 22Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

23. Assignment 2 (continued) 3.In what range of x0 can you converge to the intermediate solution? What feature of the function determines which solution is found? 23Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature