Solving 2nd order ODE with various methods

Okay the problem is of a free swinging pendulum with dampening which is modelled using the following equation:

Damping coefficient: c=1 s−1 Mass: m=1 kg Gravity: g=9.81 ms−1 Link length: l=0.5 m

We know θ(0)=90° and θ′(0)=0, solve this equation from t = 0 to t = 10 with a time interval of 0.01s

The equation is:

d2θ/dt2+(c/m)*(dθ/dt)+(g/l)*sin (θ)=0

So we need to use Euler,Heun and 4th order Runge-Kutta method

2. Relevant equations

3. The attempt at a solution

Okay so my idea was to create a function as so:

function xdot=pendemo(t,x)

% PENDEMO Pendulum ODE derivative evaluation

xdot(1,1) = x(2,1);

xdot(2,1) = -1/(1*1)*x(2,1) - 9.81/1*sin(x(1,1));

% End of pendemo.m

and than an m.file giving the above information:

xphi = [pi/2;0];

tphi = 0; 5 %start time

tfin = 10; %end time 

[t,x] = ode45('pendemo',[tphi tfin],xphi);

plot(t,x(:,1))

The only thing is how do I implement a euler/heun method? What is a 4th order Runga Kata??

thanks

Answer 1

[@Johannes]

Hello,

in your example you are using ODE45 to solve your ode. ODE45 is a solver which is using the method of Runge Kutta (RKF45) to solve ode's. This means you are using already a Runge Kutta solver of the order 4. Explicit Euler and Heun are to be implemented by you as functions. For example use the equations form wikipedia.

https://en.wikipedia.org/wiki/Euler_method

https://en.wikipedia.org/wiki/Heun%27s_method

Regards, John

[Steven]

Okay this is my code for euler:

function [t,x] = euler(dydt,tspan,y0,h)

t = (tspan(1):h:tspan(2));
n=length(t);

y = y0*ones(n,1);

for i = 1:n-1
    y(i+1)= y(i) + dydt(t(i), y(i))*(t(i+1)-t(i));

end

How can i apply it to my ODE?