Given the position and velocity vectors at one point in an orbit, one can calculate the keplerian elements and thus graph the subsequent orbital path as a circle/ellipse/parabola/hyperbola depending on the eccentricity. To simplify this problem, I am only focusing on an elliptical 2D orbit, which is defined by the semi-major axis a, eccentricity e, and argument of periapsis ω. Imagine a satellite is given a random position and velocity around a central body (such that allows for an elliptical orbit), and these elements are solved for. Graphing the projected orbital path is easy using these three orbital elements, and integrating forward in small timesteps will result in the satellite following a very close approximation of this graph.
However, increasing the speed of the simulation (by increasing the timestep) sacrifices accuracy and causes noticeable energy drift over time. My question is the following - is there a way to calculate the future position of the satellite as a function of time without integrating? Based on my research, I initially though it would be simple to calculate mean anomaly, then eccentric anomaly, and finally true anomaly, then translate this to position, but I am struggling for a few reasons; for instance, time of periapsis passage is unknown when the orbital elements are solved for. I am not an expert in this field, and I would really appreciate a breakdown of the right equations or an explanation of how to approach this problem.