Numerical aspects of time elements

Numerical results have shown that the use of time elements with time transformations provides increased accuracy in the numerical solution of gravitational systems.To gain additional accuracy improvements, it appears that the time and the time element should be calculated from quantities that have been adjusted so as to satisfy the energy integral exactly.We also have found that by reducing the growth of the time element to being nearly linear rather than quadratic causes an increase in the magnitude of the local truncation error in the solution but with a decrease in the rate of growth of the truncation error.