The finite-difference method is a method of numerical integration that
determines changes in dependent variables over finite intervals of time.
Consider an object of mass m in straight-line motion along the x-axis.
The force acting on the object may depend on time, position, and velocity.
The corresponding acceleration will also be a function of time, position, and velocity, and thus cannot easily be integrated to find velocity and position.
a = F/m = 1/m ΣFx
Suppose that at a particular time, to, the position x(to) and velocity vx(to) is known. By definition, the acceleration of the object at to is
1/m ΣFx(to, x(to),
The time derivative of vx at to can be approximated
Substituting this into Eq. 2 gives an approximate expression for the
velocity at to + Δt as
vx(to + Δt)
+ 1/m ΣFx(to, x(to),
The relationship between the velocity and position can be approximated
From this an approximate expression for the position at
time to + Δt can be determined
x(to + Δt)
= x(to) + vx(to) Δt (4)
Thus, if the position and velocity are known at
time to, then their values can be approximated at to + Δt
by using Eqs. 3 and 4. The procedure can then be repeated using x(to
+ Δt) and v(to + Δt)
as the initial conditions to determine the approximate position and
velocity at to + 2 Δt. Continuing in this
fashion, an approximate solution for the position and velocity can be obtained in terms of time.
Because of the number of calculations required, this process is greatly
simplified with the use of a computer.