Fourth Order Runge Kutta for Systems

I wrote a functions that takes in a system of first order differential equations from a file named dydt_sys and solves them. I got the code to run but my numbers are off. Any help would be appreciated.. I should get

0 0 0

2.0000 19.1656 18.7256

4.0000 71.9311 33.0995

6.0000 147.9521 42.0547

8.0000 237.5104 46.9345

10.0000 334.1626 49.4027

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% seperate m-file that holds the system. 
function dy = textbook_example_sys(t,y)
dy = [y(2); 9.81-0.25/68.1*y(2)^2];
end 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% runge-kutta system code
function [t_vals,y_vals] = RK4_sys(dydt_sys,t_span,y_0,h)
 
t_start = t_span(1); 
t_final = t_span(2); 
t_vals = (t_start:h:t_final)';
num_steps = length(t_vals);
y_vals(1,:) = y_0; 

for i=1:num_steps-1  
    
    k_1 = dydt_sys(t_vals(i),y_vals(i,:)); 
    k_2 = dydt_sys(t_vals(i)+h/2,y_vals(i,:)+k_1/2*h); 
    k_3 = dydt_sys(t_vals(i)+h/2,y_vals(i,:)+k_2/2*h); 
    k_4 = dydt_sys(t_vals(i)+h,y_vals(i,:)+k_3*h); 
    
    y_vals(i+1,:) = y_vals(i) + (k_1+2*(k_2+k_3)+k_4)/6*h; 
    
end
end 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% command line
>> t_span = [0 10];y_0 = [0 0]; h = 2; 
>> [t_vals,y_vals] = RK4_sys(@textbook_example_sys,t_span,y_0,h); 

Answer 1

[Angel Nunez]

I still kept getting an error when i did that but it got me thinking about how I was playing with column and row vectors. I ended up transpoing all of them and it fixed the problem. Thank you!

    k_1 = dydt_sys(t_vals(i),y_vals(i,:))'; 
    k_2 = dydt_sys(t_vals(i)+h/2,y_vals(i,:)+k_1/2*h)'; 
    k_3 = dydt_sys(t_vals(i)+h/2,y_vals(i,:)+k_2/2*h)'; 
    k_4 = dydt_sys(t_vals(i)+h,y_vals(i,:)+k_3*h)'; 

Answer 2

[James Tursa]

This line does not have the rhs state syntax correct:

y_vals(i+1,:) = y_vals(i) + (k_1+2*(k_2+k_3)+k_4)/6*h; 

It should be this instead

y_vals(i+1,:) = y_vals(i,:) + (k_1+2*(k_2+k_3)+k_4)/6*h; 

[Angel Nunez]

That's what i originally thought, but I got the error

Unable to perform assignment because the size of the left side is 1-by-2 and the size of the right side is 2-by-2.

Error in RK4_sys (line 16)

y_vals(i+1,:) = y_vals(i,:) + (k_1+2*(k_2+k_3)+k_4)/6*h;

[James Tursa]

You need to decide whether your state vector is going to be a row vector or a column vector and be consistent. Currently you have it mixed. E.g. your derivative function is a column vector:

dy = [y(2); 9.81-0.25/68.1*y(2)^2];

but your state update equations are row vectors:

y_vals(i+1,:) = y_vals(i,:) + (k_1+2*(k_2+k_3)+k_4)/6*h;

I would suggest going with the column vector approach to make it compatible with ode45 and friends. So change all of the y_vals(i,:) etc to y_vals(:,i) etc.

Answer 3

[Meysam Mahooti]

https://www.mathworks.com/matlabcentral/fileexchange/55430-runge-kutta-4th-order?s_tid=srchtitle

https://www.mathworks.com/matlabcentral/fileexchange/61130-runge-kutta-fehlberg-rkf78?s_tid=srchtitle