Numerical solution of ODEs

Transcription

1 Péter Nagy, Csaba Hős H-1111, Budapest, Műegyetem rkp. 3. D building. 3 rd floor Tel: Fax:

2 Table of contents Homework Introduction to Matlab programming Solution of an initial value problem (IVP) Solution of a boundary value problem (BVP)

3 Homework Homework: The homework and these slides will be available on: Submission: Only the source code and the plots have to be archived and submitted to as NC_Name.zip. The subject has to be DENUM NC Name. The deadline is the end of 12 th week. Programming language: Matlab preferred, otherwise personal submission with own laptop is necessary Free alternatives: Octave, Freemat, Scilab

4 Introduction to Matlab programming MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment Current folder! Menu Command window Workspace Text editor Errors, warnings File explorer

5 Useful commands 1. Type stg. + Tab key -> autocomplete F1 key over a function -> Help+description+syntax ; at the end of a command -> the output is not written to the command window clc -> Clear command window clear all -> Delete variables close all -> close figures Avoid overwriting predefined functions and variables (F1 or Tab)!!! Predefined variables: pi, i-imaginary unit, exp(1) Euler number function output = name(var1, var2) global var1 var2 -> define global variables, subfunctions can read them [x1, x2, x3] -> generates a row vector [x1; x2; x3] -> generates a column vector A -> the transpose of A (A can be a matrix or a vector)

6 Useful commands 2 {.+.-.*./} -> calculate {+ - * /} element by element Defining vectors linspace(a, b, n) -> generates linearly spaced vectors between a and b, n is the number of elements a:b generates linearly spaced vectors between a and b, increment is 1 a:deltax:b generates linearly spaced vectors between a and b, increment is deltax v(i) -> the ith element of the vector A(i,j) -> the element of a matrix in the row i and column j A(i,:) -> the i th row of a matrix ; A(:,j) -> the j th column of a matrix Loops for i=1:num_steps end %code %comment while (statement) end %code %comment

7 Useful commands 3 Define functions f(x,y) = Expression can be defined in Matlab f=@(x,y) Expression; -> for simple expression (eg. ODES) or function output = f (x,y) output=expression; -> for longer, more complex calculation (eg. solver) end Important functions [t, y]=ode45(@(t,y) f(y,t), t 0, x 0 ) -> solve ODE (f(y,t)); initial time: t 0 ;initial condition x 0 fminsearch=(@(y) f(y, var1, var2), y 0 ) -> minimize f(y), the initial guess is y 0 Measure computation time (t) tic %code toc (or t=toc)

8 Useful commands 4 Visualization figure(i) -> generates an empty figure, (i is an integer) subplot(n, m, i) -> generates sub figures, n: number of rows, m: num. of columns, i: is the active subplot plot(x,y) -> plot points; x and y are vectors, further possibilities-> F1 in the case of multiple plots plot(x 1,y 1, x 2,y 2, x 3,y 3 ), where x i,y i are vectors title('title') -> generates title legend('title') -> generates title xlabel('label of x axis'), ylabel('label of y axis'), xlim([xmin xmax]), ylim([ymin ymax]) saveas(figure(i), 'name.png') -> save figure(i) as name.png

9 Solution of an IVP

10 Transcript the differential equation The ODE is given 1. If the ODE is higher order transform to first order ODE System Example: t (0) 0 y= f y, ; y y x t c x t s x t g x, t ; x 0 =x ; x 0 =x 0 0 In matrix form: x : y ; x : y x : y ; 1 2 2,, y t c y t sy t g x t y t cy t sy t g x t y= y ; T 1 y2 y1= x y x0 y= t (0) 0 s c y +, ; g y2, t f y y y x 0

11 Example In the case of Van der Pool oscillator x 1 x x x 0; x 0 =x ; x 0 =x y 1 y y y y 1 y y y y y Implemented as VdP_rhs(t,y) in ExpImplEu_VanDerPol_new.m

12 Explicit Euler method 2. Select a solver Explicit Euler method Implemented as y y dt f y n 1 n n n, t

13 Implicit Euler method Implicit Euler method y y dt f y Implemented as n1 n n1 n1 Try to minimize the expression n n1 n1 n1, t y dt f y, t y 0 r y = y dt f y, t y 0 n1 n n1 n1 n1

14 Break

15 Adaptive timestep Problem: Large timestep -> accuracy, stability problems Small timestep -> computationally expensive (=slow) Idea: 1. Predict the local error -> 2. hold between given limits 1. Prediction of the local error Calculate one step in two different ways with different accuracy (but with the same scheme!) The accuracy can be changed: the order of the scheme is changed (ode45) Solve the problem with different stepsize. First, in one step with dt, then in two steps with dt/2

16 Adaptive timestep The error: 1 n1 2 n1 Hold the error between limits y y the less accurate result the more accurate result 2 1 e = y n1 y n1 min, max If the estimated error is too large Else max max do not store the results decrease the timestep dt r dt r<1 new old store the results If the est. error is too small min increase the timestep

17 Adaptive timestep Implemented as:

18 Accuracy of solvers The problem: x t x t 3sin 2t ; x 0 = 2; x 0 = 1 sin y t y t 3 2t 2 1 y1= x y2 y Analytical solution: sin cos sin x t t 2 t 2t

19 Accuracy of solvers

20 Accuracy of solvers

21 Boundary value problem x t y t v g x t 0 0 y t 0 0 y x 1 0 x t vt 0 y t x / v 0 y t 0 1?

22 BVP-Shooting method y t g y t 0 0 y t 0 Solve Implemented as: / D : y t x v D 0 1

23 BVP-Finite differences y t g y t 0 0 y t 1 / v 0 y 2y y y t 2 t yi 1 2yi yi 1 g 2 t i1 i i1 2 yi 1 2yi yi 1 t g 2 1 y y tg y n y n1 1

24 BVP-Finite differences

25 Thank you for your attention!