Problems implementing Runge Kutta to solve a Damped Pendulum - vb.net

Question

I am a high school student working on a "home project" to animate a damped pendulum by solving differential equations using the Runge Kutta method.
(The equations can be seen here: http://www.maths.tcd.ie/~smurray/Pendulumwriteup.pdf)
I have been informed that in my code, my implementation of RK4 is not correct, and to be honest I have been struggling to understand it.
The program is written in VB.net 2010.
My code for solving the equations are as follows:
Public Sub RK4Solve(ByRef The As Decimal, ByRef Ome As Decimal, ByRef h As Decimal)
l1 = h * Ome
k1 = h * f(The, Ome, h)
l2 = h * (0.5 * l1) + Ome
k2 = f((The + (0.5 * h * k1)), (Ome + (0.5 * h * l1)), (t + (0.5 * h)))
l3 = h * (0.5 * l2) + Ome
k3 = f((The + (0.5 * h * k2)), (Ome + (0.5 * h * l2)), (t + (0.5 * h)))
l4 = h * l3 + Ome
k4 = f((The + (h * k3)), (Ome + (h * l3)), (t + h))
'Setting next step of variables
The = The + (h / 6 * (l1 + (2 * l2) + (2 * l3) + l4))
Ome = Ome + (h / 6 * (k1 + (2 * k2) + (2 * k3) + k4))
t += h
End Sub
I am aware that I am multiplying each step by too many h's - I am just lost on what is happening.
My full code:
Public Class Form1
Dim l As Decimal = 1 'Length of rod (1m)
Dim g As Decimal = 9.81 'Gravity
Dim w As Decimal = 0 ' Angular Velocity
Dim initheta As Decimal = -Math.PI / 2 'Initial Theta
Dim theta As Decimal = -Math.PI / 2 'Theta (This one changes for the simulation)
Dim t As Decimal = 0 'Current time of the simulation
Dim h As Decimal = 0.01 'Time step
Dim b As Decimal = Math.Sqrt(g / l) 'Constant used in the function for dw/dt
Dim k As Decimal = 0 'Coefficient of Damping
Dim initialx = l * Math.Sin(initheta) 'Initial Amplitude of the pendulum
Private Sub Form1_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load
End Sub
'Function for dw/dt
Public Function f(ByRef the As Decimal, ByRef omega As Decimal, ByRef time As Decimal)
Return ((-b ^ 2) * Math.Sin(the)) - (k * omega) + (initheta * Math.Cos(omega * time))
End Function
Dim k1, k2, k3, k4, l1, l2, l3, l4 As Decimal 'Initialising RK4 variables
Public Sub RK4Solve(ByRef The As Decimal, ByRef Ome As Decimal, ByRef h As Decimal)
l1 = h * Ome
k1 = h * f(The, Ome, h)
l2 = h * (0.5 * l1) + Ome
k2 = f((The + (0.5 * h * k1)), (Ome + (0.5 * h * l1)), (t + (0.5 * h)))
l3 = h * (0.5 * l2) + Ome
k3 = f((The + (0.5 * h * k2)), (Ome + (0.5 * h * l2)), (t + (0.5 * h)))
l4 = h * l3 + Ome
k4 = f((The + (h * k3)), (Ome + (h * l3)), (t + h))
'Setting next step of variables
The = The + (h / 6 * (l1 + (2 * l2) + (2 * l3) + l4))
Ome = Ome + (h / 6 * (k1 + (2 * k2) + (2 * k3) + k4))
t += h
End Sub
'Timer ticking every 0.1s
'Time step is 0.01s to increase accuracy of results for testing
Private Sub Timer1_Tick(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Timer1.Tick
ComboBox1.Items.Add(theta) 'Adding theta to a drop down box to test data
RK4Solve(theta, w, h)
End Sub
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
Timer1.Enabled = False
End Sub
End Class
I have been trying to solve this for a while now, and I'm on my last legs so I am resorting to asking for help. Thanks to anyone that can!

Answer 1

It helps if you separate the differential equations from the RK4 implementation. Then you can implement RK4 as in the documentation
k1 = f1( y1, y2, x)
l1 = f2( y1, y2, x)
k2 = f1( y1 + 0.5*h*k1, y2 + 0.5*h*l1, x + 0.5*h)
l2 = f2( y1 + 0.5*h*k1, y2 + 0.5*h*l1, x + 0.5*h)
k3 = f1( y1 + 0.5*h*k2, y2 + 0.5*h*l2, x + 0.5*h)
l3 = f2( y1 + 0.5*h*k2, y2 + 0.5*h*l2, x + 0.5*h)
k4 = f1( y1 + h*k3, y2 + h*l3, x + h)
l4 = f2( y1 + h*k3, y2 + h*l3, x + h)
y1 = y1 + h/6*(k1+2*(k2+k3)+k4)
y2 = y2 + h/6*(l1+2*(l2+l3)+l4)
x = x + h
Not only does it help to avoid the duplication of the multiplication by h, but also the duplication of the addition of the base point values Ome and The.