In this article I develop a fifth order Runge Kutta Butcher.
The update formula for the fifth order Runge Kutta Butcher algorithm is shown below.
This algorithm is implemented in the Runge Kutta Butcher 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 Runge Kutta Butcher 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 RungeKutta5Butcher();
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.
The code for the Runge Kutta Butcher class is given below. The code for the Runge Kutta Base class was posted previously.
public class RungeKutta5Butcher : RungeKuttaBase
{
public RungeKutta5Butcher()
: 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[6, 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 / 4.0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + (ks[0, i] / 4.0) * currentStep;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[1, i] = rungeKuttaFunctions[i]();
}
//k2s
x.Value = x0 + currentStep / 4.0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + (ks[0, i] / 8.0 + ks[1, i] / 8.0) * currentStep;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[2, i] = rungeKuttaFunctions[i]();
}
//k3s
x.Value = x0 + currentStep / 2.0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + (-ks[1, i] / 2.0 + ks[2, i]) * currentStep;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[3, i] = rungeKuttaFunctions[i]();
}
//k4s
x.Value = x0 + currentStep * 3.0 / 4.0;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + (ks[0, i] * 3.0 / 16.0 + ks[3, i] * 9.0 / 16.0) * currentStep;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[4, i] = rungeKuttaFunctions[i]();
}
//k5s
x.Value = x0 + currentStep;
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
rungeKuttaParameters[i].Value = y0[i] + (-ks[0, i] * 3.0 / 7.0 + ks[1, i] * 2.0 / 7.0 + ks[2, i] * 12.0 / 7.0 - ks[3, i] * 12.0 / 7.0 + ks[4, i] * 8.0 / 7.0) * currentStep;
}
for (int i = 0; i < rungeKuttaParameters.Length; i++)
{
ks[5, 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] + ((7.0 * ks[0, i] + 32.0 * ks[2, i] + 12.0 * ks[3, i] + 32.0 * ks[4, i] + 7.0 * ks[5, i]) / 90.0) * currentStep;
row[i + 1] = rungeKuttaParameters[i];
}
returnCollection.Add(row);
}
return returnCollection.ToArray();
}
}
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