Numerical Differentiation and Quadrature (Integration) 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 Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Numerical Differentiation Problem: Though an analytical derivative can be found for all differentiable functions, it is often impractical to calculate it Solution: Approximate the derivative numerically Method of forward finite differences: Remember that: Therefore: for small h Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Numerical Quadrature (Integration) Problem: Generally, it is not possible to find the antiderivative (Stammfunktion) of a function f(x) in order to solve a definite integral in the interval [a,b] Solution: Approximate the area under the curve numerically Trapezoidal rule: Divide the interval into sub-intervals and approximate the integral by the sum of the areas of the resulting trapezoids Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Trapezoidal rule a x1 x2 xn-1 b Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

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

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 Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

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 Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

How does Matlab do it? quad: Low accuracy, non-smooth integrands, uses adaptive recursive Simpson rule quadl: High accuracy, smooth integrands, uses adaptive Gauss/Lobatto rule (degree of integration routine related to number of points) quadgk: High accuracy, oscillatory integrands, can handle infinite intervals and singularities at the end points, uses Gauss/Konrod rule (re-uses lower degree results for higher degree approximations) Degree q of an integration rule = Polynomials up to order q can be integrated accurately (assuming there is no numerical error) Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Matlab Syntax Hints All the integrators use the syntax result = quadl(int_fun, a, b, ...); int_fun is a function handle to a function that takes one input x and returns the function value at x; it must be vectorized Use parametrizing functions to pass more arguments to int_fun if needed f_new = @(x)int_fun(x, K); f_new is now a function that takes only x as input and returns the value that int_fun would return when K is used as second input Note that K must be defined in this command This can be done directly in the integrator call: result = quadl(@(x)int_fun(x, K), a, b); Matlab 2012a and newer: integral(...) Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Assignment 1 Consider the function Use the method of forward finite differences to approximate the derivative of f(x) at x = 1. Vary h between 10-15 and 10-1 using logspace(-15, -1, 200), and calculate the error of the finite differences approximation compared to the analytical solution for each h. Plot the error vs. h using loglog. What do you observe? What could be the cause for this behavior? Repeat the calculations of 1. and 2. using the method of centered finite differences. Compare the two loglog plots. Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Exercise Mass Transfer into Semi-Infinite Slab c(z, t) Consider a liquid diffusing into a solid material The liquid concentration at the interface is constant The material block is considered to be infinitely long, the concentration at infinity is therefore constant and equal to the starting concentration inside the block c(z, t) z c0 = const. c∞ = const. Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Exercise (continued) Using a local mass balance, we can formulate an ODE where j is the diffusive flux in [kg m-2 s-1] With Δz  0, we arrive at a PDE in two variables z z+Δz jin jout Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Exercise (continued) By combining this local mass balance with Fick’s law, a PDE in one variable is found: The analytical solution of this equation (found by combination of variables) reads: Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Assignment 2 Write a Matlab program to calculate and plot the concentration profile in the slab Use the following values: c∞ = 0; c0 = 1 Create a vector zeta = linspace(1e-6, 3), calculate the value of c for each zeta, then plot c vs. zeta Use integral or quadl for the integration Create a function which calculates an integral with the trapezoidal rule Use the form: function F = trapInt(f, n, a, b) Where f is a function handle to the function that is to be integrated, n is the number of points and a and b denote the interval Provide the mean error of the two methods (1. and 2.) using fprintf Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature

Assignment 2 (continued) Improve your function by adding a convergence check: In addition to computing the integral with n points, simultaneously calculate it with 2n points while the results differ by more than 10-6, double n and iterate the calculation Terminate the calculation and issue a warning if n exceeds 106 Michael Sokolov / Numerical Methods for Chemical Engineers / Numerical Quadrature