Numerical solution of ODEs - Budapest University of ...

H-1111, Budapest, Műegyetem rkp. 3. D building. 3 rd floor

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Numerical solution of ODEs

Péter Nagy, Csaba Hős

2015.

Table of contents


Homework

Introduction to Matlab programming

Solution of an initial value problem (IVP)

Solution of a boundary value problem (BVP)

Homework


Homework:

The homework and these slides will be available on:

http://www.hds.bme.hu/mota/eng/denum/index.html

Submission:

Only the source code and the plots have to be archived and submitted to [email protected] as NC_Name.zip.

The subject has to be „DENUM NC Name”.

The deadline is the end of 12th week.

Programming language:

Matlab preferred, otherwise personal submission with own laptop is necessary

Free alternatives: Octave, Freemat, Scilab

http://www.hds.bme.hu/mota/eng/denum/index.html

mailto:[email protected]





Introduction to Matlab programming


MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment

File explorer

Menu Command

window

Workspace Text editor

Current folder ! Errors,

warnings

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)

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 ith row of a matrix ; A(:,j) -> the jth column of a matrix

Loops

for i=1:num_steps

%code %comment

end

while (statement)

%code %comment

end

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) , t0, x0) -> solve ODE (f(y,t)); initial time: t0 ;initial condition x0

fminsearch=(@(y) f(y, var1, var2) , y0) -> minimize f(y), the initial guess is y0

Measure computation time (t)

tic

%code

toc (or t=toc)

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(x1,y1, x2,y2, x3,y3), where xi,yi 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


Solution of an IVP

Transcript the differential equation


The ODE is given

1. If the ODE is higher order transform to first order ODE System

Example:

In matrix form:

0 0x t cx t s x t g x t 0 0, ; x =x ; x =x

1 2 2x : y x : y x : y; ;

2 2 1

2 2 1

y t c y t sy t g x t

y t cy t sy t g x t

,

,

1 2y x y=

0t (0)= , ; y f y y y

0

0

2 0

00 1t (0)

g y ts c

x= , ;

, x

y y + f y y y

T

1 2y y= ;y

Example


In the case of Van der Pool oscillator

Implemented as „VdP_rhs(t,y)” in „ExpImplEu_VanDerPol_new.m”

2

0 0

2

2 1 2 1

2

2 1 2 1

1 2

x 1 x x x 0 0 0

y 1 y y y 0

y 1 y y y

y y

; x =x ; x =x

Explicit Euler method


2. Select a solver

– Explicit Euler method

Implemented as

n 1 n n ndt t, y y f y

Implicit Euler method


– Implicit Euler method

Try to minimize the expression

Implemented as

n 1 n n 1 n 1

n n 1 n 1 n 1

dt t

dt t 0

,

,

y y f y

y f y y

n 1 n n 1 n 1 n 1dt t 0r , y = y f y y


Break

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

Adaptive timestep


The error:

Hold the error between limits

If the estimated error is too large do not store the results

decrease the timestep

Else store the results

If the est. error is too small increase the timestep

1

n 1

2

n 1

the less accurate result

the more accurate result

y

y

2 1

n 1 n 1e = y y

min max,

max

new olddt rdt r <1

min

max

Adaptive timestep


Implemented as:

Accuracy of solvers


The problem:

x t x t 3 2t 0 2 0 1sin ; x = ; x =

2 1y t y t 3 2tsin

1 2y x y=

0

2

1

y

Analytical solution:

x t t 2 t 2tsin cos sin

Accuracy of solvers


Boundary value problem


x t v

y t g

x t 0 0

y t 0 0

y x 1 0

1

x t vt 0

y t x v 0

y t 0

/

?

BVP-Shooting method


y t g

y t 0 0

y t 0

1D : y t x v/

D 0 Solve

Implemented as:

BVP-Finite differences


y t g

y t 0 0

y t 1 v 0/

i 1 i i 1

2

i 1 i i 1

2

2

i 1 i i 1

y 2y yy t

t

y 2y yg

t

y 2y y t g

2

3

2

n 2

n 1

y2 1 1

y1 2 1

t g

y1 2 1 1

y1 2 1

BVP-Finite differences



Thank you for your attention!

Numerical solution of ODEs - Budapest University of ...

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