NUMERICAL ERROR OF TOTAL-ENERGY - DEPENDENCE ON TIMESTEP

The numerical error of the total energy E of a conservative dynamical system is shown to obey dE/dt = C-k(Delta t)(k')x Sigma(n)E(n) omega(n )(k'+1), where omega(n) and E-n are the characteristic frequency and e nergy of the nth mode, which are constant or time dependent in linear or nonlinear problems, respectively. The integer k' in the above formu la is equal to k or k+1 according to k odd or even, respectively, wher e k is the order of accuracy of the integration scheme. Specifically, the commonly used Runge-Kutta 4th order scheme yields 5th order accura cy for the total energy. This behavior is not influenced whether the c oordinates and momenta behave chaotic or not.