Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.
The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order Runge-Kutta method, so I have the system:
$$ \left\{\begin{array}{l} \frac{dy}{dx} = z \\ \frac{dz}{dx} = 6y - z \end{array}\right. $$ With $y(0)=3$ and $z(0)=1$.
Now I know that for two general 1st order ODE's $$ \frac{dy}{dx} = f(x,y,z) \\ \frac{dz}{dx}=g(x,y,z)$$ The 4th order Runge-Kutta formula's for a system of 2 ODE's are: $$ y_{i+1}=y_i + \frac{1}{6}(k_0+2k_1+2k_2+k_3) \\ z_{i+1}=z_i + \frac{1}{6}(l_0+2l_1+2l_2+l_3) $$ Where $$k_0 = hf(x_i,y_i,z_i) \\ k_1 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0) \\ k_2 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1) \\ k_3 = hf(x_i+h,y_i+k_2,z_i+l_2) $$ and $$l_0 = hg(x_i,y_i,z_i) \\ l_1 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0) \\ l_2 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1) \\ l_3 = hg(x_i+h,y_i+k_2,z_i+l_2)$$
My problem is I am struggling to apply this method to my system of ODE's so that I can program a method that can solve any system of 2 first order ODE's using the formulas above, I would like for someone to please run through one step of the method, so I can understand it better.
Solution 1:
I will outline the process and you can fill in the calculations.
We have our system as:
$$ \left\{\begin{array}{l} \frac{dy}{dx} = z \\ \frac{dz}{dx} = 6y - z \end{array}\right. $$ With $y(0)=3$ and $z(0)=1$.
We must do the calculations in a certain order as there are dependencies between the numerical calculations. This order is:
- $k_0 = hf(x_i,y_i,z_i)$
$l_0 = hg(x_i,y_i,z_i)$
$k_1 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0)$
$l_1 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0)$
$k_2 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1)$
$l_2 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1)$
$k_3 = hf(x_i+h,y_i+k_2,z_i+l_2)$
$l_3 = hg(x_i+h,y_i+k_2,z_i+l_2)$
$y_{i+1}=y_i + \frac{1}{6}(k_0+2k_1+2k_2+k_3)$
- $z_{i+1}=z_i + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$
We typically need some inputs for the algorithm:
- A range that we want to do the calculations over: $a \le t \le b$, lets use $a = 0, b = 1$.
- The number of steps $N$, say $N = 10$.
- The steps size $h = \dfrac{b-a}{N} = \dfrac{1}{10}$
The system we are solving is:
$$ \frac{dy}{dx} = f(x,y,z) = z \\ \frac{dz}{dx}=g(x,y,z) = 6y - z$$
Doing the calculations using the above order for the first time step $i= 0, t_0 = 0 = x_0$, yields:
- $k_0 = hf(x_0,y_0,z_0) = \dfrac{1}{10}(z_0) = \dfrac{1}{10}(1) = \dfrac{1}{10}$
$l_0 = hg(x_0,y_0,z_0) = \dfrac{1}{10}(6y_0 - z_0) = \dfrac{1}{10}(6 \times 3 - 1) = \dfrac{1}{10}(17)$
$k_1 = hf(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_0,z_0+\frac{1}{2}l_0) = \dfrac{1}{10}(1 + \dfrac{1}{2}\dfrac{1}{10}(17)) ~~$(You please continue the calcs.)
$l_1 = hg(x_0+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_0+\frac{1}{2}l_0)$
$k_2 = hf(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_1,z_0+\frac{1}{2}l_1)$
$l_2 = hg(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_1,z_0+\frac{1}{2}l_1)$
$k_3 = hf(x_0+h,y_0+k_2,z_0+l_2)$
$l_3 = hg(x_0+h,y_0+k_2,z_0+l_2)$
$y_{1}=y_0 + \frac{1}{6}(k_0+2k_1+2k_2+k_3)$
- $z_{1}=z_0 + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$
You now have $x_1$ and $z_1$ which you need for the next time step after all of the intermediate (in order again). Now, we move on to the next time step $i = 1, t_1 = t_0 + h = \dfrac{1}{10} = x_1$, so we have:
- $k_0 = hf(x_1,y_1,z_1) = \dfrac{1}{10}(z_1)$
$l_0 = hg(x_1,y_1,z_1) = \dfrac{1}{10}(6y_1 - z_1)$
$k_1 = hf(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_0,z_1+\frac{1}{2}l_0)$
$l_1 = hg(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_0,z_1+\frac{1}{2}l_0)$
$k_2 = hf(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_1,z_1+\frac{1}{2}l_1)$
$l_2 = hg(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_1,z_1+\frac{1}{2}l_1)$
$k_3 = hf(x_1+h,y_1+k_2,z_1+l_2)$
$l_3 = hg(x_1+h,y_1+k_2,z_1+l_2)$
$y_{2}=y_1 + \frac{1}{6}(k_0+2k_1+2k_2+k_3)$
- $z_{2}=z_1 + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$
Continue this for $10$ time steps. Your final result should match closely (assuming the numerical algorithm is stable for this problem) to the exact solution. You will compare $z_{10}$ to the exact result. The exact solution is:
$$y(x) = e^{-3 x}+2 e^{2 x}$$
If we find $y(1) = \dfrac{1}{e^3} + 2 e^2 = 14.8278992662291643974401973...$.
Solution 2:
Although this answer contains the same content as Amzoti's answer, I think it's worthwhile to see it another way.
In general consider if you had $m$ first-order ODE's (after appropriate decomposition). The system looks like
\begin{align*} \frac{d y_1}{d x} &= f_1(x, y_1, \ldots, y_m) \\ \frac{d y_2}{d x} &= f_2(x, y_1, \ldots, y_m) \\ &\,\,\,\vdots\\ \frac{d y_m}{d x} &= f_m(x, y_1, \ldots, y_m) \\ \end{align*}
Define the vectors $\vec{Y} = (y_1, \ldots, y_m)$ and $\vec{f} = (f_1, \ldots, f_m)$, then we can write the system as
$$\frac{d}{dx} \vec{Y} = \vec{f}(x,\vec{Y})$$
Now we can generalize the RK method by defining \begin{align*} \vec{k}_1 &= h\vec{f}\left(x_n,\vec{Y}(x_n)\right)\\ \vec{k}_2 &= h\vec{f}\left(x_n + \tfrac{1}{2}h,\vec{Y}(x_n) + \tfrac{1}{2}\vec{k}_1\right)\\ \vec{k}_3 &= h\vec{f}\left(x_n + \tfrac{1}{2}h,\vec{Y}(x_n) + \tfrac{1}{2}\vec{k}_2\right)\\ \vec{k}_4 &= h\vec{f}\left(x_n + h, \vec{Y}(x_n) + \vec{k}_3\right) \end{align*}
and the solutions are then given by $$\vec{Y}(x_{n+1}) = \vec{Y}(x_n) + \tfrac{1}{6}\left(\vec{k}_1 + 2\vec{k}_2 + 2\vec{k}_3 + \vec{k}_4\right)$$
with $m$ initial conditions specified by $\vec{Y}(x_0)$. When writing a code to implement this one can simply use arrays, and write a function to compute $\vec{f}(x,\vec{Y})$
For the example provided, we have $\vec{Y} = (y,z)$ and $\vec{f} = (z, 6y-z)$. Here's an example in Fortran90:
program RK4
implicit none
integer , parameter :: dp = kind(0.d0)
integer , parameter :: m = 2 ! order of ODE
real(dp) :: Y(m)
real(dp) :: a, b, x, h
integer :: N, i
! Number of steps
N = 10
! initial x
a = 0
x = a
! final x
b = 1
! step size
h = (b-a)/N
! initial conditions
Y(1) = 3 ! y(0)
Y(2) = 1 ! y'(0)
! iterate N times
do i = 1,N
Y = iterate(x, Y)
x = x + h
end do
print*, Y
contains
! function f computes the vector f
function f(x, Yvec) result (fvec)
real(dp) :: x
real(dp) :: Yvec(m), fvec(m)
fvec(1) = Yvec(2) !z
fvec(2) = 6*Yvec(1) - Yvec(2) !6y-z
end function
! function iterate computes Y(t_n+1)
function iterate(x, Y_n) result (Y_nplus1)
real(dp) :: x
real(dp) :: Y_n(m), Y_nplus1(m)
real(dp) :: k1(m), k2(m), k3(m), k4(m)
k1 = h*f(x, Y_n)
k2 = h*f(x + h/2, Y_n + k1/2)
k3 = h*f(x + h/2, Y_n + k2/2)
k4 = h*f(x + h, Y_n + k3)
Y_nplus1 = Y_n + (k1 + 2*k2 + 2*k3 + k4)/6
end function
end program
This can be applied to any set of $m$ first order ODE's, just change m in the code and change the function f to whatever is appropriate for the system of interest. Running this code as-is yields
$ 14.827578509968953 \qquad 29.406156886687729$
The first value is $y(1)$, the second $z(1)$, correct to the third decimal point with only ten steps.
Solution 3:
A Matlab implementation is given below:
% It calculates ODE using Runge-Kutta 4th order method
% Author Ido Schwartz
% Originally available form: http://www.mathworks.com/matlabcentral/fileexchange/29851-runge-kutta-4th-order-ode/content/Runge_Kutta_4.m
% Edited by Amin A. Mohammed, for 2 ODEs(April 2016)
clc; % Clears the screen
clear all;
h=0.1; % step size
x = 0:h:1; % Calculates upto y(1)
y = zeros(1,length(x));
z = zeros(1,length(x));
y(1) = 3; % initial condition
z(1) = 1; % initial condition
% F_xy = @(t,r) 3.*exp(-t)-0.4*r; % change the function as you desire
F_xyz = @(x,y,z) z; % change the function as you desire
G_xyz = @(x,y,z) 6*y-z;
for i=1:(length(x)-1) % calculation loop
k_1 = F_xyz(x(i),y(i),z(i));
L_1 = G_xyz(x(i),y(i),z(i));
k_2 = F_xyz(x(i)+0.5*h,y(i)+0.5*h*k_1,z(i)+0.5*h*L_1);
L_2 = G_xyz(x(i)+0.5*h,y(i)+0.5*h*k_1,z(i)+0.5*h*L_1);
k_3 = F_xyz((x(i)+0.5*h),(y(i)+0.5*h*k_2),(z(i)+0.5*h*L_2));
L_3 = G_xyz((x(i)+0.5*h),(y(i)+0.5*h*k_2),(z(i)+0.5*h*L_2));
k_4 = F_xyz((x(i)+h),(y(i)+k_3*h),(z(i)+L_3*h)); % Corrected
L_4 = G_xyz((x(i)+h),(y(i)+k_3*h),(z(i)+L_3*h));
y(i+1) = y(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h; % main equation
z(i+1) = z(i) + (1/6)*(L_1+2*L_2+2*L_3+L_4)*h; % main equation
end