Example of the Euler Method in 2D

We will use the Euler method to solve the damped pendulum equation

which we first rewrite as a system

for initial conditions

, and

Set up initial conditions and parameters

theta0=3;
y0=0;
gamma=0.1;

Set up time stepping

h = 1/100;
tFinal=20;
nSteps=tFinal/h;
t=linspace(0,tFinal,nSteps+1);

Initialize θ & y

theta=zeros(size(t));
y = zeros(size(t));
theta(1)=theta0; y(1)=y0;

Loop through and run the Euler method. Note that this code uses a function (defined at the bottom of this file) to evaluate the right-hand side.

for k=1:nSteps
    [delta_x, delta_y] = pendulumODE(theta(k),y(k),gamma);
    theta(k+1) = theta(k) + h * delta_x;
    y(k+1) = y(k) + h * delta_y;
end

Plot the results

plot(t,theta,t,y);

xlabel('t'); legend('\theta','y')

Plot the solution in the phase plane

plot(theta,y)

xlabel('\theta');ylabel('y');

Function used to evaluate the differential equation

function [delta_x, delta_y] = pendulumODE(x,y,gamma)
delta_x = y;
delta_y = -gamma*y-sin(x);
end