The following is a sample javascript implementation of the Rünge-Kutta method of numerically solving initial value problems in ordinary differential equations. In particular, this webpage implements the so-called RK4 method which is a 4th order method.
Example
Consider the equation y'=f(x,y)=yx2 - 6*y/5. Given an initial value y(0)=y0, we want to know the approximate value of the function y(x) when x=x1.
Enter values for y0 and x1 and a number of iterations of RK4 to use to approximate y1=y(x1).
<script type="text/javascript">
var g2 =[];
function f(x,y)
{
// return the value of y'
return Math.pow(x,2)*y - 6*y/5;
}
function g(x)
{
// the is the
return Math.exp(-6*x/5+Math.pow(x,3)/3);
}
function rk4iter(x,y,h)
{
// compute the four RK coefficients using the RHS of the diff eq.
k1 = f(x,y);
k2 = f(x + h/2, y + k1 * h / 2);
k3 = f(x + h/2, y + k2 * h / 2);
k4 = f(x + h, y + k3 * h);
// compute the y value at the end of the iteration
return y + (k1 + 2*k2 + 2*k3 + k4) * h / 6;
}
function rk4(n,y0,x1)
{
var h = x1/n; // compute the step size along the x-axis
var x = 0; // initial value of x is 0
var y = (+y0); // initial value of y is y0
g2 = [[x,y]]; // start with initial conditions
// Prepare the output for reporting
var msg = '<table border="0.5"><tr><td width="150">x</td><td width="200">y</td></tr>';
msg += '<tr><td>'+x+'</td><td>'+y+'</td></tr>';
// loop until x reaches its target value
for (var i = 1; i <= n; i++)
{
// Perform iterations of the RK4 algorithm
y = rk4iter(x,y,h);
x += h;
// record the x and y values for this iteration
msg += '<tr><td>'+x+'</td><td>'+y+'</td></tr>';
// Add data to the plot as well.
g2.push([x,y]);
}
// record the actual target value, the absolute and percent error
msg += '<tr><td>Actual f('+x1+')=</td><td>'+g(x1)+'</td></tr>';
msg += '<tr><td>Absolute Error</td><td>'+Math.abs(g(x1)-y)+'</td></tr>';
msg += '<tr><td>Percent Error</td><td>'+Math.abs(Math.round((g(x1)-y)/g(x1)*1000000))/10000+'%</td></tr>';
// report the target value to the user
document.getElementById("msg").innerHTML=msg+'</table>';
// Refresh the plot
$(function () {
var d1 = [];
for (var i = 0; i <= 40; i++)
d1.push([i/20, g(i/20)]);
$.plot($("#placeholder"), [ { label: "actual", data: d1} , { label: "RK4", data: g2, points: {show: true}}]);
});
}
</script>