Algorithm for solving electromagnetic problems using only forces Is there any fundamental issue to solving electromagnetic problems with the following algorithm? (practicality aside)
i) Set position, velocity, mass and charge for a set of particles.
ii) Compute the electric field at the position of every particle produced by all the other particles with Coulomb law.
iii) Compute the magnetic field at the position of every particle produced by all the other particles with Biot-Savart law.
iv) Move all the particles a differential amount using Newton's second Law with Lorentz Force:
for every particle i compute: $m \vec a = q(\vec E + \vec v \times \vec B)$
v) Go to step ii.
 A: Yes. At least two that I can see offhand:

*

*Coulomb's law only holds in Electrostatics, meaning it does not hold true for moving charges, even those moving with a uniform velocity with respect to each other. This is because the electric field for a moving charge is no longer the "usual" $1/r^2$ electric field as you can see in Chapter 26 of the Feynman Lectures (see Fig 26-4).


*The Biot-Savart law similarly only holds for Magnetostatics, where you deal with steady currents. A single moving point charge is certainly not a steady current!
Furthermore, since these fields aren't constant, you should also remember that changes in the electromagnetic field travel at the speed of light $c$. In other words, the charges will not sense an instantaneous force as you describe, but a retarded one, retarded by a time $t - r/c$ where $r$ is the distance between the charges.
Now, you could do a little bit better by actually using the exact electric and magnetic fields of moving charges (these are derived in the chapter of the Feynman Lectures I linked above), taking into account the retardation,  and then use he formula:
$$\mathbf{F} = q (\mathbf{E + v \times B}),$$
but I also see a fourth problem: Accelerated charges radiate energy in the form of electromagnetic waves. This emission causes a recoil force on the charged particle called the Abraham-Lorentz (or radiation reaction) force. You'd need to take this into account as well for a complete description. However this too is only valid at speeds that are small compared to the speed of light $c$. Its relativistic version is the Abraham-Lorentz-Dirac force, I believe.
But this sounds like a very complicated problem without making some assumptions first (taking the non-relativistic limit, etc.).
A: I think I the very first page of Fenyman lectures Vol 2. mentions this when it tries to motivate why we use fields and not just forces.
https://www.feynmanlectures.caltech.edu/
It's a bit sparse but:
https://www.feynmanlectures.caltech.edu/II_01.html

It turns out that the forms in which the laws of electrodynamics are simplest are not what you might expect. It is not simplest to give a formula for the force that one charge produces on another. It is true that when charges are standing still the Coulomb force law is simple, but when charges are moving about the relations are complicated by delays in time and by the effects of acceleration, among others. As a result, we do not wish to present electrodynamics only through the force laws between charges; we find it more convenient to consider another point of view—a point of view in which the laws of electrodynamics appear to be the most easily manageable.

So I guess the answer to your question is that your method does not take into account the finite speed at which disturbances in the magnetic and electric fields propagate between charges.
