13.3. Runge–Kutta Fourth-Order Method (RK4)

Try these first

1. 1.

   What is the order of accuracy for the RK4 method?

   Hint

   Think about how many evaluations are made.

2. 2.

   What does RK4 stand for?

   Hint

   Connect the abbreviation to its full name.

3. 3.

   What method uses four calculations for slope estimation?

   • Euler's Method
   • Runge-Kutta Method
   • Trapezoidal Rule

   Hint

   Consider the number of evaluations each method employs.

4. 4.

   True or False: The RK4 method always requires less computational effort than RK2.

   • True
   • False

   Hint

   Reflect on the workload involved with each method.

5. 5.

   Given the differential equation dy/dx = sin(x) with y(0) = 0, apply the RK4 method over the interval [0, π/2] with a step size of h = 0.1. Calculate y(π/2).

   Hint

   Pay close attention to how the sin function behaves within this interval.

6. 6.

   Consider the equation dy/dx = (y^2)/x with the initial condition y(1) = 1. Use RK4 to approximate y(2) using h = 0.5.

   Hint

   Use each k value carefully as y grows rapidly in this region.