Numerical error of total energy. Dependence on timestep Original Research Article

The numerical error of the total energy E of a conservative dynamical system is shown to obey dE/dt=Ck(Δt)kʹX ∑nEnωKʹ+1n, where ωn and En are the characteristic frequency and energy of the nth mode, which are constant or time dependent in linear or nonlinear problems, respectively. The integer k′ in the above formula is equal to k or k + 1 according to k odd or even, respectively, where k is the order of accuracy of the integration scheme. Specifically, the commonly used Runge-Kutta 4th order scheme yields 5th order accuracy for the total energy. This behavior is not influenced whether the coordinates and momenta behave chaotic or not.