Using Improved Euler Method in Matlab - matlab

Question

I am trying to solve a 2nd order differential equation in Matlab. I was able to do this using the forward Euler method, but since this requires quite a small time step to get accurate results I have looked into some other options. More specifically the Improved Euler method (Heun's method).
I understand the principle of Improved Euler method, that it first estimates the velocity and then uses that information to correct it to the current condition. But I am not totally sure if what I have written is totally correct.
1)Can you check if my code utilizes the Improved Euler method correctly?
2)In my code, the last line before the end, the second B(ii) should be B(ii+1)?
I have written a simplified code for both options. Here it is:
t = 0:0.01:100;
dt = t(2)-t(1); % Time step
%Constants%
M = 20000;
m_a = 10000;
c= 15000;
k_spring = 40000;
B = rand(1,length(t)+1);
%% Forward Euler Method %%
x = zeros(1,length(t)+1); % Pre-allocation
u = zeros(1,length(t)+1); % Pre-allocation
x(1) = 1; % Initial condition
u(1) = 0; % Initial condition
for ii = 1:length(t)
x(ii+1) = x(ii) + dt*u(ii);
u(ii+1) = u(ii) + dt * ((1/(M+m_a)) * -(c+k_spring+B(ii))*x(ii));
end
%% Improved Euler Method %%
x1 = zeros(1,length(t)+1); % Pre-allocation
u1 = zeros(1,length(t)+1); % Pre-allocation
x1(1) = 1; % Initial condition
u1(1) = 0; % Initial condition
for ii = 1:length(t)
x1(ii+1) = x1(ii) + dt*u1(ii);
u1(ii+1) = u1(ii) + dt * ((1/(M+m_a)) * -(c+k_spring+B(ii))*x1(ii)); %Estimate
u1(ii+1) = u1(ii) + (dt/2) * ( ((1/(M+m_a)) * -(c+k_spring+B(ii))*x1(ii)) + ((1/(M+m_a)) * -(c+k_spring+B(ii))*x1(ii+1)) ); %Correction
end
Thanks!

Answer 1

You should follow the principal programming idea to make things that are used repeatedly into extra procedures. Suppose you did so and the function evaluating the ODE function is called odefunc.
function [dotx, dotu] = odefunc(x,u,B)
dotx = u;
dotu =(1/(M+m_a)) * -(c+k_spring+B)*x);
end
Then
for ii = 1:length(t)
%% Predictor
[dotx1,dotu1] = odefunc(x1(ii), u1(ii), B(ii));
%% One corrector step
[dotx2,dotu2] = odefunc(x1(ii)+dotx1*dt, u1(ii)+dotu1*dt, B(ii+1));
x1(ii+1) = x1(ii) + 0.5*(dotx1+dotx2)*dt;
u1(ii+1) = u1(ii) + 0.5*(dotu1+dotu2)*dt;
end