In numerical analysis, the Runge–Kutta methods are a family of implicit and explicit iterative methods, which include the well-known routine called the Euler Method, used in temporal discretization for the approximate solutions of ordinary differential equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Martin Kutta. Here, integration of the normalized two-body problem from t0 = 0 [s] to t = 3600 [s] for an eccentricity of e = 0.1 is implemented and compared with analytical method.
Goddard Trajectory Determination System (GTDS): Mathematical Theory, Goddard Space Flight Center, 1989.
Meysam Mahooti (2020). Runge Kutta 8th Order Integration (https://www.mathworks.com/matlabcentral/fileexchange/55431-runge-kutta-8th-order-integration), MATLAB Central File Exchange. Retrieved .
I'm afraid that the screenshot that you posted contains a mistake. In the reference that you provided (https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017203.pdf) page 288, the first sum in index seven is -231, whereas you screenshot shows -234
The formula is taken from the following link:
Long A. C., Cappellari J. O., Velez C. E., Fuchs A. J.; Mathematical Theory of the Goddard Trajectory Determination System; Goddard Space Flight Center; FDD/552-89/001; Greenbelt, Maryland (1989).
How it can be true if the minimum number of stages s required for an explicit s-stage Runge–Kutta method to have order 8 is 11? There is also the result obtained by Butcher (1985) that "For p≥8 no explicit Runge-Kutta method exists of order p with s = p+2 stages"
Hairer, 2008, 2nd edition, p. 179
Image changed with a higher quality.
test_RK8.m is modified.
Accuracy assessment is added to RK8_test.m