# Simulating electrostatics in discrete time steps

I am trying to simulate the motion of several charged particles that are free to move around but have repulsive forces between each other. These may be 10 electrons moving around causing an instantaneous Newtonian force on all other particles (not accounting for relativistic effects).

I want to run the simulation at discrete time steps (i.e. every 10ms). Is there any way to find the position and velocity of each particle at after each time step?

Originally, I thought I would just calculate each force at the beginning of the time step and then use these forces to calculate the positions and velocities for the next time step. The problem is that all particles have moved during the time step, so my assumption of a constant force acting on the particle during the time step will be wrong.

I could make the time steps really small so that the displacement of the other particles is nearly zero to approximate a stationary force. This would require more computations as well as still induce small errors.

Am I stuck with having to approximate this with error is there a way to compute and integral over each time step to find the next positions and velocities?