In this article I develop a second order Runge Kutta Improved Polygon class.
The update formula for the Runge Kutta Improved Polygon algorithm is shown below.
This algorithm is implemented in the second order Runge Kutta Improved Polygon class, which is given at the bottom of this post.
As an example of using this class, I will use the following equations.
x1’ = x1 – 2x2
x2’ = 2x1 + x2
Where x1(0) = 0, x2(0) = 4
x1 = -4etsin(2t)
x2 = 4etcos(2t)
To solve these equations using the second order Runge Kutta Improved Polygon class, with a step size of 0.1 from 0.0 to 3.3, the code below is used.
Parameter T = new Parameter(0.0);
Parameter X1 = new Parameter(0.0);
Parameter X2 = new Parameter(4.0);
Parameter[] parameters = new Parameter[] { X1, X2 };
Func<double>[] functions = new Func<double>[]
{
() => X1 - 2.0 * X2,
() => 2.0 * X1 + X2
};
RungeKuttaBase ode = new RungeKutta2ImprovedPolygon();
double[][] dRes = ode.Integrate(parameters, T, functions, 3.3, 0.1);
A graph of these results are given below.
As can be seen, the second order Runge Kutta Improved Polygon class gives much more accurate results than the Euler class with a step size of 0.1.
The code for the second order Runge Kutta Improved Polygon class is given below. The code for the Rung Kutta Base class was posted previously.
public class RungeKutta2ImprovedPolygon : RungeKuttaBase
{
public RungeKutta2ImprovedPolygon()
: base()
{
}
public override double[][] Integrate(Parameter[] rungeKuttaParameters, Parameter x, Func<double>[] rungeKuttaFunctions, double xEnd, double step)
{
Debug.Assert(rungeKuttaParameters.Length == rungeKuttaFunctions.Length);
double xStart = x;
double stepSize = (xEnd - xStart) / step;
int integerStepSize = (int)stepSize;
if (stepSize > (double)integerStepSize) { integerStepSize++; }
integerStepSize++;
Collection<double[]> returnCollection = new Collection<double[]>();
double[] row = new double[rungeKuttaParameters.Length + 1];
row[0] = x;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
row[i + 1] = rungeKuttaParameters[i];
}
returnCollection.Add(row);
double currentX = xStart;
double[,] ks = new double[2, rungeKuttaParameters.Length];
double x0 = xStart;
double[] y0 = new double[rungeKuttaParameters.Length];
double currentStep = step;
while (currentX < xEnd)
{
if (currentX + currentStep > xEnd) { currentStep = xEnd - currentX; }
x0 = currentX;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
y0[i] = rungeKuttaParameters[i];
}
//k0s
x.Value = x0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[0, i] = rungeKuttaFunctions[i]();
}
//k1s
x.Value = x0 + currentStep / 2.0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + ks[0, i] * currentStep / 2.0;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[1, i] = rungeKuttaFunctions[i]();
}
//Final
currentX += currentStep;
row = new double[rungeKuttaParameters.Length + 1];
row[0] = currentX;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + ks[1, i] * currentStep;
row[i + 1] = rungeKuttaParameters[i];
}
returnCollection.Add(row);
}
return returnCollection.ToArray();
}
}
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