Runge-Kutta Methods

We created a differential equation to test the RK4 method by starting with the solution

y(t)=(1/t)sin²(t)

and differentiating to obtain

y'=-(1/t²)sin²(t)+(2/t)sin(t)cos(t)

y'= -y/t + 2 sqrt(y/t)cos(t)

We wrote the program rk4.c to approximate y(3) from the initial y(1)=sin²(1) using the RK4 method.

Output from the program for the RK4 method was

y(3)=   6.638285563573941e-03
Y3=   6.638285558272329e-03
E=   5.301611580299515e-12

To compare RK4 with Euler's explicit method we changed the main loop to read as

    for(int i=0;i<n;i++){
        double t=t0+i*h;
        y+=h*f(t,y);
    }

Output from the modified program using Euler's method was

y(3)=   6.524827660108781e-03
Y3=   6.638285558272329e-03
E=  -1.134578981635487e-04

While RK4 is more complex and takes 4 function evaluations per times step the error is much smaller than Euler's method. A fair comparision, in terms of the number of function evaluations of f, could be made by allowing Euler's method to take 4 times as many steps. In this case Euler's method produces

y(3)=   6.609982887022521e-03
Y3=   6.638285558272329e-03
E=  -2.830267124980836e-05

which is still much less accurate than the RK4 method. It is possible to make a more advanced analysis of the error in the RK4 method by making a log-log plot of how E depends on n.

Last Updated: Fri Oct 24 12:11:24 PDT 2014