A numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time.
Attempts to create such a model established the more general field of celestial mechanics.
The results of this simulation can be compared with past measurements to check for accuracy and then be used to predict future positions.
The former are easier, but extremely calculation intensive, and only practical on an electronic computer.
Strictly speaking, the latter was not much less calculation intensive, but it was possible to start with some simple approximations and then to add perturbations, as much as needed to reach the wanted accuracy.
In essence this mathematical simulation of the Solar System is a form of the N-body problem.
The symbol N represents the number of bodies, which can grow quite large if one includes the Sun, 8 planets, dozens of moons, and countless planetoids, comets and so forth.
The result for each planet is an orbit, a simple description of its position as function of time.
Once this is solved the influences moons and planets have on each other are added as small corrections.
Although this method is no longer used for simulations, it is still useful to find an approximate ephemeris as one can take the relatively simple main solution, perhaps add a few of the largest perturbations, and arrive without too much effort at the wanted planetary position.
One starts with a high accuracy value for the position (x, y, z) and the velocity (vx, vy, vz) for each of the bodies involved.
When also the mass of each body is known, the acceleration (ax, ay, az) can be calculated from Newton's Law of Gravitation.
Next one chooses a small time-step Δt and applies Newton's Second Law of Motion.
The velocity multiplied with Δt gives a correction to the position.
Repeating this procedure often enough, and one ends up with a description of the positions of all bodies over time.
The advantage of this method is that for a computer it is a very easy job to do, and it yields highly accurate results for all bodies at the same time, doing away with the complex and difficult procedures for determining perturbations.
The disadvantage is that one must start with highly accurate figures in the first place, or the results will drift away from the reality in time; that one gets x, y, z positions which are often first to be transformed into more practical ecliptical or equatorial coordinates before they can be used; and that it is an all or nothing approach.
In reality this is not the case, except when one takes Δt so small that the number of steps to be taken would be prohibitively high.
Computers cannot integrate, they cannot work with infinitesimal values, so instead of dt we use Δt and bringing the resulting variable to the left:
The simplest way to solve these is just the Euler algorithm, which in essence is the linear addition described above.
Limiting ourselves to 1 dimension only in some general computer language: As in essence the acceleration used for the whole duration of the timestep, is the one as it was in the beginning of the timestep, this simple method has no high accuracy.
For example, for distances in the Solar System the astronomical unit is most straightforward.
If this is not done one is almost certain to see a simulation abandoned in the middle of a calculation on a floating point overflow or underflow, and if not that bad, still accuracy is likely to get lost due to truncation errors.
The total amount of energy and angular momentum of a closed system are conserved quantities.
By calculating these amounts after every time step the simulation can be programmed to increase the stepsize Δt if they do not change significantly, and to reduce it if they start to do so.
In the case of comets, nongravitational forces, such as radiation pressure and gas drag, must be taken into account.
In the case of Mercury, and other planets for long term calculations, relativistic effects cannot be ignored.
The finite speed of light also makes it important to allow for light-time effects, both classical and relativistic.
For example, the flattening of the Earth causes precession, which causes the axial tilt to change, which affects the long-term movements of all planets.
Long term models, going beyond a few tens of millions of years, are not possible due to the lack of stability of the Solar System.