2. This system of two second-order differential equations can be simplified to a system of four first-order differential equations by introducing two

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1 UPPSALA UNIVERSITET Inst. för informationsteknologi Avd. för teknisk databehandling Jonas Nilsson Tekniska beräkningar EI1 Miniprojekt: ODE De uppgifter som beskrivs nedan har karaktären av (förhoppningsvis inspirerande) undersökning. Betoningen ligger mindre på programmering och mera på experimenterande, med syftet att du skall fördjupa din förståelse av de numeriska problemställningarna. Uppgiften är hämtad ur kompendiet Notes on Numerical Computing (av M. Overton, C. Paige och M. Thuné) och därför formulerad på engelska. Frågeställning This question concerns the motion of a satellite around the sun: the satellite could be a planet, a comet or an unpowered spacecraft, whose mass is assumed to be negligible compared with the mass of the sun. The motion of such a satellite takes place in a plane, so we can describe its coordinates by just two space dimensions x and y. Assume that the sun is at the origin, (0; 0). The system of differential equations describing the satellite's motion is Newton's law of gravitation, discovered by Newton in the late 17th century, namely x 00 x = k ( px 2 + y 2 ) 3 y 00 y = k ( px 2 + y 2 ) 3 : This shows how the acceleration of the satellite depends on its position with respect to the sun. The constant k depends on the choice of distance units that we make: let us assume that the units of distance are chosen so that k =1. Uppgifter 1. Suppose you double the distance the satellite is from the sun, that is you double both x and y. Does the acceleration x 00 and y 00 increase or decrease, and by what factor? The answer to this question explains why Newton's law of gravitation is called an inverse square law. 1

2 2. This system of two second-order differential equations can be simplified to a system of four first-order differential equations by introducing two new variables z = x 0 (t) (the velocity, or speed, of the satellite in the x space direction) and w = y 0 (t) (the velocity of the satellite in the y space direction). Write down this system of equations. (This trick is exactly the same as the trick for reducing the second-order differential equation for the mass suspended on a spring to a system of two first-order differential equations.) 3. Let us solve these differential equations in Matlab. Matlab does include tools for solving differential equations (type lookfor differential). One of these are used in 8. Here is the file satell.m, which we shall use as the basis for our solution. 1 % satell.m: 2 % plot the path of a satellite (a planet or a comet) 3 % in orbit around the Sun 4 % m-files needed: slope.m, euler.m 5 6 axis([ ]); % this is a good choice of axis 7 axis('square'); % makes circles look like circles. 8 plot(0,0,'g*'); % plot position of the sun 9 hold on; % hold plot for overwriting 10 t = 0; % initialize time to zero 11 x(1) = 4; % initial x space coordinate is 4 12 x(2) = 0; % initial y space coordinate is 0 13 x(3) = 0; % initial velocity in x direction is zero 14 x(4) = input('enter x(4), the velocity in y direction '); 15 tmax = input('enter tmax, the total simulated time '); 16 h = input('enter h, the the time step size '); 17 plot(x(1),x(2),'k*'); % plot initial (x,y) point 18 while t < tmax, 19 euler; % script file to take an Euler step 20 % WRITE A SCRIPT FILE FOR MIDPOINT EULER TOO: 21 % IT'S MUCH BETTER 22 plot(x(1),x(2),'k*'); % plot the new (x,y) point 23 hold on; 24 end; 25 hold off; % so subsequent plots will start fresh This script does most of the necessary work: read it carefully. It uses a 2

3 vector x to store the four variables x; y; z; w in x(1), x(2), x(3), x(4) respectively: thus the first two elements of x are the space coordinates x,y and the last two are the velocities x 0, y 0. The initial values are chosen to be: x =4, y =0, x 0 =0, and y 0, the velocity inthey direction, to be input by you, the user. The script file also prompts you for h, (the time step size) and tmax (the amount of simulated time to run the computation). The script needs two more Matlab files. Here is the first: % euler.m: script file to take one step of Euler's method % requires x ( a vector) and t, h (scalars) to be given x = x + h*slope(t,x); t = t + h; % x is a vector, so is slope(t,x) % t is a scalar (time) The second file needed by satell.m is slope.m, a function file which you must write. The first line should be function s = slope(t,x), and it should assign values to s(1),..., s(4) which are the values of the derivatives of the four variables x(1),...x(4) in terms of the current values x(1),...,x(4), using the formulas you worked out to answer the previous question. (The variable t should be ignored by the slope function since the derivatives depend only on x, not on t.) Using the input parameters mentioned above, the satell script should plot some sort of approximate orbit. If you make h too small, the program will take too long to run, but if you make h too large, it will give a very inaccurate orbit, so you need to experiment with h. You also need to experiment with the initial value of y 0 = x(4), the velocity in the y direction: if this is too large, the satellite will fly off into outer space, but if the velocity is too small, the satellite will fall into the sun. Experiment until you find values which take the satellite around the sun in an orbit which is approximately a circle, ending up approximately where it started. The satellite will not return exactly to where it started, because Euler's method is not very accurate. If you are not able to find initial values for which the satellite goes around the sun, you probably have the equations specified wrong in slope.m. 4. To make the orbit close up properly, with the satellite returning where it started, you could either reduce h or use midpoint Euler instead of Euler's method. The program takes a long time to run if h is too small, so write a script file mideuler.m which performs one step of midpoint Euler instead of Euler's method. Note that euler.m is extremely simple since it operates on the vector of four variables all at once: midpoint Euler is not much 3

4 more complicated. Once you have done this correctly, you will find that satell.m plots a nearly perfect orbit. 5. Now experiment with values of y 0 which are a little larger than the value which gives you a circular orbit. You should see the satellite traveling along the path of an ellipse. The sun is at one of the two focal points of the ellipse. The fact that planets and comets have elliptical orbits was observed by Kepler in the early 17th century, but it was not explained until Newton developed his law of gravitation 60 years later. The eccentricity " of the ellipse can be defined as " p 1» 2, where» large diameter=small diameter. Thus " = 0 corresponds to» = 1, a circle (having zero eccentricity ). As " % 1, the ellipse becomes more elongated (eccentric). From the more general description of a conic section, " = 1corresponds to a parabolic path, and " >1 ahyperbolic path. The eccentricity of the ellipse depends on the size of the initial velocity y 0. The orbits of planets, such as the Earth, are typically not very eccentric, in fact they look almost like circles. But the orbits of comets, such as Halley's comet, can be very eccentric. That is why Halley's comet can be seen from the Earth only once every 76 years: the rest of the time it is very far from the sun traveling around its highly eccentric orbit. 6. Now experiment with larger values for y 0. If you make the initial value for y 0 large enough, the orbit becomes a hyperbola, with the satellite approaching the direction of a straight line heading right out of the solar system; in this case the satellite escapes the gravitational attraction of the sun. Many comets pass through the solar system on hyperbolic orbits: they enter the solar system, pass by the sun once and then leave the solar system again. Only one spacecraft launched from Earth has ever left the solar system on a hyperbolic orbit: that is Voyager II, which was launched in the late 1970's, flew by Uranus in 1986, and is now on the way out of the solar system, never to return. The borderline case between elliptical and hyperbolic orbits is when y 0 is exactly equal to the escape velocity: the value just large enough for the satellite to escape from the solar system. In this case the orbit is actually a parabola, which is the borderline case" between an ellipse and a hyperbola. (In Jules Verne's book, From Earth to Moon, two characters are arguing about whether they are on a parabolic or a hyperbolic path, while the third, who is less scientific, is horrified to realize that they are doomed in either case.) 7. Now experiment with smaller initial values for y 0. You will find that as you decrease y 0 below the value giving the circular orbit, the orbit again 4

5 becomes an ellipse, with smaller values for y 0 taking the satellite closer to the sun. If you make the initial value for y 0 too small, the satellite passes so close to the sun that the discrete approximation becomes unstable and the satellite appears to swing off out of the solar system. This is a result of discretization, not a physical behavior. (If you make h small enough you can eliminate this instability, but it will take too long to wait for the plot. A better idea would be to implement a more accurate method, see 8.) 8. Now compare your previous results with MATLAB:s ODE-solver ode45. Change row in satell.m to: 16 % solve the ode for the time t <= tmax 17 [T,X] = ode45( 'slope', [t, tmax], x ); 18 % plot the path of the satellite 19 plot( X(:,1), X(:,2), 'k*' ); In order to make MATLAB:s ODE-solver ode45 to work properly, you have to change the function slope to return a column vector, i.e. add the following line s = s'; in the end of the file slope.m. Extra uppgift för dig som blivit inspirerad Write an adaptive Euler's method for solving a differential equation. The idea is similar to that in the adaptive integration computer assignment, except that it cannot be implemented recursively, since the differential equation is solved starting at t =0and continuing from there. But the idea of estimating the error is the same. You try an Euler step with size h, and two Euler steps with size h=2, and compare the results. The difference in x is the error estimate. Suppose we want the error estimate to be approximately ffl. If it is larger than ffl, then reduce the step size h by a factor of 2 and try again. Once the error estimate is smaller than (or equal to) ffl, the method can go on to the next step. But, if the error is smaller than ffl=10, then increase h by 2forthenext step, since we don't want h to get unnecessarily small. Use this to solve the gravitation equations with the same conditions as given before, for two different values of the initial y velocity: ffl One which is close to the ellipse hyperbola borderline. This will give you a really big ellipse so you can't get very close to the borderline value! 5

6 ffl One which gives you an eccentric ellipse, where the satellite passes close to the sun, and for which the ordinary Euler method is unstable. How eccentric an ellipse can you solve? Redovisning Arbetet skall redovisas i samband med det avslutande seminariet. Alla grupper skall redovisa, även om man inte lyckats lösa uppgifterna helt. Redovisningen består av två delar, en skriftlig och en muntlig. Normalt kommer två till tre grupper att få redovisa muntligt vid varje seminarium. Schema för de muntliga redovisningarna utdelas separat. Den skriftliga redovisningen, som alla grupper skall lämna, skall innehålla följande punkter med lämpliga rubriker: ffl En sammanfattande problembeskrivning om minst 7 10 rader. ffl Beskrivning av hur gruppen löst problemet. (Denna beskrivning får inte utgöras av programtexten, utan skall bestå av en förklaring av tillvägagångssätt och resonemang.) ffl Redovisning av resultat (grafer, tabeller, eller vad som kan vara lämpligt för den aktuella uppgiften). ffl Gruppens kommentarer till resultaten. Försök om möjligt ge teoretiska förklaringar till dem. Peka också på sådant som gruppen tycker verkar konstigt, intressant, värt att få närmare belyst, etc. ffl Om gruppen inte lyckats lösa uppgifterna skall man redovisa det arbete som ändå utförts och särskilt förklara vad det var man körde fast på. ffl Programlistor bifogas sist som bilagor. 6