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"Ordinary Differential Equations"
RungeKutta-Fehlberg Method
Results for a coupled spring system with Step Size h

t
x1
x2
v1
v2

Solution from RungeKutta-Fehlberg Method

Step Size h = { 0. },




IMPLEMENTATION
Runge Kutta-Fehlberg Method

As with solving a single first-order ordinary differential equation, this method is a more efficient and effective method for performing numerical integration, because it can estimate the truncation error at each integration step and automatically adjust the step size to keep the error within a prescribed tolerance.

Error Algorithm Creation

There is simply no single choice that works well in all problems. To control the largest component of E(h), that error measure would be:

error(h) = MAXi|Ei(h)|

To control some gross measure of the error, the root-main-square error defined by

error(h) = √ 1/n ∑i=0n-1 E2i (h)

where n is the number of first-order differential equations.



Testing the Runge Kutta-Fehlberg Method

To test the method as defined above, a new TestRungaKuttaFehlberg() static method has been added and executed. Supporting code and methods are not shown.

An initial "Step Size h = { 0.5}" for the test has been defined.

           static void TestRunge KuttaFehlberg();
              {
                 ListBox1.Items.Clear();
                 ListBox2.Items.Clear();
                 ListBox3.Items.Clear();
                 ListBox4.Items.Clear();
                 ListBox5.Items.Clear();
                 double dt = t1;
                 double t0 = 0.0;
                 VectorR x0 = new VectorR(new double[] { x10, x20, v10, v20 });
                 VectorR x = x0;
                  for (int i = 0; i < 21; i++);
                 {
                  double t = 0.1 * i;
                   x = ODE.MultiRungeKuttaFehlberg(f1, t0, x, t, dt, 1e-5);
                   ListBox1.Items.Add(" " + t );
                   ListBox2.Items.Add(" " + x[0]);
                   ListBox3.Items.Add(" " + x[1]);
                   ListBox4.Items.Add(" " + x[2]);
                   ListBox5.Items.Add(" " + x[3]);
                 }
              }

The user can manipulate step size and try variations (0.1 - 0.9) only.



Other Implementations...


Object-Oriented Implementation
Graphics and Animation
Sample Applications
Ore Extraction Optimization
Vectors and Matrices
Complex Numbers and Functions
Ordinary Differential Equations - Euler Method
Ordinary Differential Equations 2nd-Order Runge-Kutta
Ordinary Differential Equations 4th-Order Runge-Kutta
Higher Order Differential Equations
Nonlinear Systems
Numerical Integration
Numerical Differentiation
Function Evaluation


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