Naive numerical approximations

When an analytical solution to a problem is not available, then one must resort to numerical methods for solving Newton's equations. In the average numerical methods book, some methods that appear there are

1.

Runge-Kutta methods.

2.

Predictor-corrector methods.

3.

Euler's method.

4.

Verlet (and variant) methods.

Of these, only the last is suitable for use with Newton's equations. The reason for this will be made clear later when we study the fundamental symmetries of Newton's equations and then require the any numerical method respect those symmetries. Most available methods do not.

Once again, we shall illustrate the problem of numerical solution by considering a single particle moving in one dimension. This will be easily generalizable to N particles moving in three dimensions. In the one-dimensional case, we have to solve

subject to the initial conditions x(0), v(0). The solution will be uniquely given as functions of t and x(0) and v(0):

such that f(0;x(0),v(0)) = x(0) and g(0;x(0),v(0)) = v(0) and . An obvious approach would be simply to perform a Taylor expansion of the unknown functions f and g about t=0,

Recognizing that , we can write the position equation as

Now since we do not have any information about the system beyond the second derivative (Newton's second law tells nothing about , for example), we need to truncate the expansion at the second order term. Thus, it will only be valid for times small enough that the term can be considered negligible. Let us introduce a small time interval and write

which tells us that the solution will be good up to second order in .

Unfortunately, if we look at the velocities, we actually need to truncate this expansion at the first order, since we know nothing about . If we truncate v(t) at first order, however, we have

which is only good to first order in .

It would be far better if we could find an approximation for v(t) that was good to second order in just as the position is. In fact, this is possible if we only use the information in the position equation. To this end, we seek an expression for . This can be obtained by taking the position equation

where, although ``='' is used, it must be remembered that the solution is now an approximate solution, not the true solution. Let us now consider starting from and applying the rule backwards in time (i.e. for a time ) so that we end up back at x(0). The rule would be

Now, if we solve this for , we find

Using the rule for , we have so that

The combination of the two rules gives a numerical algorithm:

which is known as the velocity Verlet algorithm. Note that both the position and velocity are good to second order in . It is one of the most widely used algorithms for solving Newton's equations numerically. It will be shown later that the algorithm respects all of the fundamental symmetries of Newton's equation.

Note that it is straightforward to generalize the algorithm for N particles moving in 3 dimensions:

where the shorthand notation means and means .

Some comments are in order at this point:

1.

Note that the algorithm can be iterated. Given x(0) and v(0), we use the rules once to obtain . Then, given these, we can use the rules again to obtain according to

and the procedure can be repeated to yield , thereby generating a solution of total time at discrete time points, , ,..., .

2.

The overall error in a solution of n iterations of the rules will not actually be the same as the error in a single iteration (which is good to second order in ). To see this, note that the overall error after n iterations will be

Thus, the overall error in a solution of total time t will be good only to first order in with a prefactor that grows as t. It is important to keep this in mind when quoting time-step errors in numerical solutions.

3.

Other popular numerical solution methods that are equivalent or nearly equivalent to the velocity Verlet method exist. One of these is the so called Verlet method given by

Note that the velocities are not used here, explicitly, and must be obtained by a finite difference formula

This method is exactly equivalent to velocity Verlet.

Another populat method is the so called position Verlet method given by

We will see later how algorithms such as these are derived in a more formal and powerful way. It will also be shown there that the position Verlet method is not exactly equivalent to the velocity Verlet method, although it is also an algorithm that is good to second order in .

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