Lecture on Numerical Analysis
9
M. Dumbser 1 / 9 Analisi Numerica Università degli Studi di Trento Dipartimento d‘Ingegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser Lecture on Numerical Analysis Dr.-Ing. Michael Dumbser 17 / 12 / 2007
description
Lecture on Numerical Analysis. Dr.-Ing. Michael Dumbser. 17 / 12 / 2007. Ordinary Differential Equations. An equation of the form. is called a nonlinear ordinary differential equation (ODE) of first order. The simplest first order ODE is the equation. with. - PowerPoint PPT Presentation
Transcript of Lecture on Numerical Analysis
M. Dumbser
1 / 9
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Lecture on Numerical Analysis
Dr.-Ing. Michael Dumbser
17 / 12 / 2007
M. Dumbser
2 / 9
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Ordinary Differential Equations
)),(( ttyfdt
dy
An equation of the form
is called a nonlinear ordinary differential equation (ODE) of first order. The simplest first order ODE is the equation
aydt
dy
The exact solution of this simple ODE is obtained as follows:
adty
dy
)(tyy with
t
t
y
y
adty
dy
00
ttyy aty
00ln
00lnln ttayy 00
ln ttay
y 0
0)( ttaeyty
M. Dumbser
3 / 9
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Ordinary Differential Equations
A simple nonlinear ordinary differential equation (ODE) of first order is given by
2aydt
dy
The exact solution of this simple nonlinear ODE can be computed in a similar way:
adty
dy
2
)(tyy with
t
t
y
y
adty
dy
00
2 tt
y
y
aty 0
0
1
00
11tta
yy
1
00
1)(
tta
yty
M. Dumbser
4 / 9
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Numerical Methods for the Integration of ODE
The simplest numerical method for the integration of first order ODEs is theso-called explicit (forward) Euler method, invented already in the 18th century by the mathematician Leonhard Euler. It corresponds to the use of a first order accuratefinite difference scheme to approximate the derivative on the left hand side of the ODEand to compute the operator on the right hand side at the current state yn and time tn:
),( tyfdt
dy
),(1
nnnn
tyft
yy
),(1 nnnn tyftyy
Computing the function f on the right hand side at the unknown time tn+1, weobtain the so-called implicit (backward) Euler method:
),( 111 nnnn tyftyy
ttt nn 1
M. Dumbser
5 / 9
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Numerical Methods for the Integration of ODE
A simple modification of the forward Euler scheme, which yields a second order accurate method, is based on the following approach: Integrate the ODE from time level tn to time level tn+1:
),( 21
211 nnnn tyftyy
Unfortunately, the value of y is not known at tn+1/2, but we can approximate it making a half time-step of the forward Euler scheme. This means, that first, a so-called predictor is computed at the half time-step t/2:
),(2
21 nnnn tyf
tyy
11
),(
n
n
n
n
t
t
t
t
dttyfdt
dy
1
),(1
n
n
t
t
nn dttyfyy
The integral on the right hand side is now evaluated using a one-point Gaussian quadrature formula (the midpoint is the integration point):
),( 21
211 nnnn tyftyy
The corrector is then:
M. Dumbser
6 / 9
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Numerical Methods for the Integration of ODE
One of the most famous and mostly commonly used schemes for ODE integration is the explicit fourth-order scheme of Runge and Kutta:
43211 22
6kkkk
tyy nn
),(1nn tyfk
),( 2122tntn tkyfk
),( 2223tntn tkyfk
),( 34 tttkyfk nn
M. Dumbser
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Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Exercise 1Write a MATLAB script ODECompare.m that integrates the following two ODE:
(1) Use the explicit (forward) Euler method, using different time steps and compare with the exact solutions.
(2) Use the modified second order Euler method (=second order Runge-Kutta scheme) with different time steps, and compare with the exact solutions.
(3) Use the fourth order Runge-Kutta scheme with various time steps. Compare with the exact solutions.
ydt
dy 2y
dt
dy 1)0( y ]10;0[t
M. Dumbser
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Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Exercise 2Wite a MATLAB script Springs.m that integrates the linear ODE system for position
and velocity for a system of N point masses, connected by springs as follows:
m1mi-1 mi
mi+1 mN
The ODE system for the positions of the point masses is simply:
ii vdt
dx
The forces acting on point mass i are
iiiii kLxxF
1 111 iiiii kLxxF
Where ki are the spring stiffnesses and Li are the lengths of the springs in their
undeformed state. The ODE system for the velocities of the point masses is then
iii
i FFmdt
dv 1
01 F0
NF
Use the second order Runge-Kutta scheme (modified Euler method).
M. Dumbser
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Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Exercise 3Wite a MATLAB script Orbit.m that integrates the linear ODE system for position
and velocity for a planet, turning around a star.
The ODE system for the position vector of the planet is:
vdt
xd
The gravitation force of the star, acting on the planet, is given by:
2
PS
PS
PS
PSG
xx
mm
xx
xxF
The ODE system for the velocity vector of the planet is
GP
P Fmdt
vd 1
Use the second order Runge-Kutta time integration scheme (modified Euler method).
