The update formula for the fourth order Runge Kutta algorithm is shown below.
This algorithm is implemented in the fourth order Runge Kutta 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 fourth order Runge Kutta 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 RungeKutta4();
double[][] dRes = ode.Integrate(parameters, T, functions, 3.3, 0.1);
Step size of 0.1.
Step size of 0.25.
Step size of 0.5.
Step size of 1.0.
As can be seen, the fourth order Runge Kutta class gives much more accurate results than the third order Runge Kutta class.
The code for the fourth order Runge Kutta class is given below. The code for the Runge Kutta Base class was posted previously.
public class RungeKutta4 : RungeKuttaBase
{
public RungeKutta4()
: 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[4, 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]();
}
//k2s
x.Value = x0 + currentStep / 2.0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + ks[1, i] * currentStep / 2.0;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[2, i] = rungeKuttaFunctions[i]();
}
//k3s
x.Value = x0 + currentStep;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + ks[2, i] * currentStep;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[3, 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[0, i] + 2.0 * ks[1, i] + 2.0 * ks[2, i] + ks[3, i]) / 6.0) * currentStep;
row[i + 1] = rungeKuttaParameters[i];
}
returnCollection.Add(row);
}
return returnCollection.ToArray();
}
}
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