Comparison of methods

While I did not analyze the performance of the above methods using different software, hardware, or even trying different ODE, I will present results from the examples I illustrated earlier. The fastest of three methods described above was Runge-Kutta, Euler method was slightly slower ( about 2% ), and Taylor series methods was slower by about 4%. I think the Runge-Kutta method performed the best, although it has the most formulae, because the formulae themselves are more optimizable by javac, the Java compiler. This is an important point, because it may be possible with the release of Java Hotspot server to achieve better performance than directly compiled code, such as C/C++.

Euler's method had the biggest error: for all the iterations it accumulated to about 0.4. Taylor series produced an error of 1.0E-3, and Runge-Kutta 5.5E-11.

Taking performance, error, and flexibility Runge-Kutta method looks the most promising. For more computationally demanding feats there exist few types of adaptive Runge-Kutta methods (e.g. Dormand-Prince method), which optimize performance without significant increase in error.