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 ΣF_{x}
(t,x,v_{x}) (1)
Suppose that at a particular time, t_{o}, the position x(t_{o}) and velocity v_{x}(t_{o}) is known. By definition, the acceleration of the object at t_{o} is
=
1/m ΣF_{x}(t_{o}, x(t_{o}),
v_{x}(t_{o})) (2)
The time derivative of v_{x} at t_{o} can be approximated
as
Substituting this into Eq. 2 gives an approximate expression for the
velocity at t_{o} + Δt as
v_{x}(t_{o} + Δt)
= v_{x}(t_{o})
+ 1/m ΣF_{x}(t_{o}, x(t_{o}),
v_{x}(t_{o}))Δt (3)
The relationship between the velocity and position can be approximated
as
From this an approximate expression for the position at
time t_{o} + Δt can be determined
as
x(t_{o} + Δt)
= x(t_{o}) + v_{x}(t_{o}) Δt (4)
Thus, if the position and velocity are known at
time t_{o}, then their values can be approximated at t_{o} + Δt
by using Eqs. 3 and 4. The procedure can then be repeated using x(t_{o}
+ Δt) and v(t_{o} + Δt)
as the initial conditions to determine the approximate position and
velocity at t_{o} + 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. |