# Fourth Order Runge Kutta for Systems

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Angel Nunez on 18 Apr 2019
Answered: Meysam Mahooti on 5 May 2021
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);
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### Accepted Answer

Angel Nunez on 18 Apr 2019
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)';
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### More Answers (2)

James Tursa on 18 Apr 2019
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;
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James Tursa on 18 Apr 2019
Edited: James Tursa on 18 Apr 2019
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.

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Meysam Mahooti on 5 May 2021
https://www.mathworks.com/matlabcentral/fileexchange/61130-runge-kutta-fehlberg-rkf78?s_tid=srchtitle
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