To accurately determine the trajectory of bodies in near-earth space, various numerical methods are used to solve the Cauchy problem of ordinary differential equations. Common methods today are multistep implicit Gauss — Jackson method of 8 order of accuracy , explicit one-step embedded Dormand — Prince methods  and Kutta — Felberg , as well as collocation one-step implicit Gauss — Everhart methods .The first method has a significant drawback characteristic of all multistep methods — the difficulty of changing the integration step and finding multiple solutions at the initial time. The methods of Dormand — Prince, like any explicit methods of Runge — Kutta, have a rather high staging, which increases their resource intensity. Gauss collocation methods are devoid of these disadvantages. They combine high accuracy with relatively low stages and flexibility of changing the integration step.
Unresolved to date are questions about the estimation of the local error and the choice of a step in the Everhart method. The well-established algorithm described in  greatly overestimates the integration error, which leads to an unreasonable decreasing the step, and as a consequence, to an increasing the calculation time. The authors of this paper proposed another algorithm based on the Runge rule. Testing has shown that for near-Earth orbits, the acceleration of integration when using it reaches up to 2 times (for elliptical orbits) compared to the well-known analogue with the same accuracy of the result. In addition, the new method for estimating the error and choosing the step has shown good properties in the integration of the Arenstorf orbit
The authors also investigated the use of equinoctial elements  for integrating the equations of motion of the spacecraft by Everhart methods. Equinoctial elements, with their proper definition , do not have singularities, and contain only one parameter, which changes significantly during the movement of the body. These circumstances make them extremely convenient for finding trajectories. The authors compared the integration rate in Cartesian coordinates and equinoctial elements for the perturbed problem. Asphericity of the gravitational potential (expansion of the potential into a Gaussian series with coefficients from the EGM2008 model), aerodynamic drag (NRLMSISE-00 atmospheric model, approximation of free molecular hypothermic flow) and solar radiation pressure with a cone shadow model were considered as perturbations. When using equinoctial elements to integrate the equations of translational motion by Gauss — Evehart methods, it was found that the process of finding a solution can be accelerated by 20% (for low-orbit spacecraft) and more (for high orbits).