Matlab Euler & #39; s method, Matlab eulermethod

Matlab Euler's method, Matlab eulermethod

% matlab script to test efficiency of% Euler's method, classical Runge-Kutta, and ode45% on Arenstorf orbit problemclose allclear all% these are variables we would like the right-hand function to "see"% without actually passing them as argumentsglobal mu muHat% set normalized massesmu = 0.012277471;muHat = 1 - mu;% set time span of integrationt0 = 0;tf = 17.1;% set initial conditionsu1 = 0.994;u2 = 0;u1Dot = 0;u2Dot = -2.00158510637908252240537862224;% pack into vector y0y0 = [u1 u2 u1Dot u2Dot]';% set the name of the function to compute the right-hand sidef = 'arenstorf';disp(['The ' f ' problem'])disp(' ')% begin with Euler's methoddisp('Experiments with Euler''s method.')% "logical" variable to end given phase (Euler, Runge-Kutta)iMore = 1;while (iMore == 1),    % prompt user for number of steps    nSteps = input('Please enter the number of steps to take: ');    % set up array to store solution    y = zeros(length(y0),nSteps+1);    h = (tf-t0)/nSteps;    t = linspace(t0,tf,nSteps+1);    y(:,1) = y0;    tic    for i=1:nSteps,        y(:,i+1) = y(:,i) + h*feval(f,t(i),y(:,i));    end    toc    plot(y(1,:),y(2,:))    iMore = input('Do you wish to repeat with Euler''s method? (1=yes,0=no) ');    disp(' ')end    close all% now investigate classical Runge-Kutta disp('Experiments with classical Runge-Kutta method.')iMore = 1;while (iMore == 1),    % prompt user for number of steps    nSteps = input('Please enter the number of steps to take: ');    % set up array to store solution    y = zeros(length(y0),nSteps+1);    h = (tf-t0)/nSteps;    t = linspace(t0,tf,nSteps+1);    y(:,1) = y0;    tic    for i=1:nSteps,        K1 = feval(f,t(i),y(:,i));        K2 = feval(f,t(i)+h/2,y(:,i)+h*K1/2);        K3 = feval(f,t(i)+h/2,y(:,i)+h*K2/2);        K4 = feval(f,t(i+1),y(:,i)+h*K3);        y(:,i+1) = y(:,i) + h*(K1+2*(K2+K3)+K4)/6;    end    toc    plot(y(1,:),y(2,:))    iMore = input('Do you wish to repeat with classical RK? (1=yes,0=no) ');    disp(' ')endclose all% finally use ode45disp('Solving problem with ode45 ... ')% try with and without options in call to ode45options = odeset('RelTol',1e-4);tic[t,y] = ode45(f,[t0 tf],y0,options);tocplot(y(:,1),y(:,2))% from Wikipedia about the Arenstorf orbit% Richard F. Arenstorf is an American mathematician who discovered a% stable orbit between the Earth and the Moon, called an Arenstorf% Orbit, which was the basis of the orbit used by the Apollo Program% for going to the Moon.% The Arenstorf Orbit% While the orbit of a satellite around the Sun was empirically% discovered by Kepler and theoretically proven by Newton to be an% ellipse, at the time when the United States was interested in going% to the Moon, there was no such solution known for the shape of a% satellite orbiting regularly around two objects, such as a% spacecraft going between the Earth and the Moon. This is a special% case of the infamous Three Body Problem, for which a general% analytical solution is not known because of its complexity of% solving the effect of three bodies which all pull on each other% while moving, a total of six interactions. However the case of an% Earth-Moon satellite can be simplified to four interactions, because% although the three objects gravitationally all pull on each other,% the effect of the spacecraft's gravity upon the motion of the vastly% more massive Earth and Moon is practically non-existent. Arenstorf% found a stable orbit for a spacecraft orbiting between the Earth and% Moon, shaped like an '8' with the Earth or Moon located inside each% loop of the '8'. This orbit is the basis of a path going to the Moon% from the Earth, such as the United States Apollo program. For a% permanent presence on the Moon, it would be the path of what% Arenstorf calls a 'Space Bus', a ferry which could regularly orbit% supplies and people between the Earth and Moon without directly% expending fuel. By staying on the Arenstorf orbit, lunar astronauts% automatically return back to Earth. Before leaving NASA at the first% Moon Landing, Arenstorf mapped out an emergency rescue orbit, which% was used in the Apollo 13 incident, in which a catastrophic% malfunction forced aborting the Moon landing, but the astronauts% ultimately returned safely to Earth without a major course% adjustment.