  |
|
|
|
In a model problem consisting of a relatively large system of
distinct objects, such as the planets example we've been
developing over the last few weeks, the real action comes
in solving Newton's equations of motion. As a quick reminder,
Newton's equations say that dx/dt = v, and dv/dt = F/m, where
x is the position of a particle, v is its velocity, F is the
force acting on it, and m is its mass. In this notation,
dx/dt means the change of x with respect to time, or how
much the position changes in a given period of time.
To do this summation and solve Newton's partial differential equations, it would be necessary to integrate both sides of the expressions. Exact integration over all possible values of x, t, and v is usually not possible; and, thus, Newton's equations can't be solved exactly. Because of this fact, some approximations must be made to use these equations in a meaningful simulation. (If you're interested in learning more about some of these terminologies, it would probably be helpful to seek out information on calculus and differential equations). Not being able to solve Newton's equations exactly means that we will have to pick a time step, or dt, that is not infinitessimal. As it turns out, the accuracy of a result using such an approximation depends on the size of the time step chosen. Because we want to simulate as much physical time as possible, we need to choose a method that will allow us to use a relatively large time step while conserving the total energy of the system to be examined (conservation of energy is the most commonly used measure of accuracy in these types of simulations). Almost without exception, the methods of choice for integrating Newton's equations of motion derive from the Verlet integrator. From a computational standpoint, the Verlet method offers speed of calculation, low memory burdens and the use of a relatively large time step. Some of these considerations will be discussed in later articles. The most prevalent Verlet integrators are the Verlet Leapfrog Integrator, the Verlet Velocity Integrator, the ABM4 Integrator and the Runge-Kutta-4-Integrator. For our purposes in developing the planets problem, we will consider the velocity version of the Verlet algorithm. Next week, we'll begin to look at some of the details that need to be fleshed out before the planetary system Go To Page: 1 2
The copyright of the article Integrating Newton's Equations in Scientific Computing is owned by . Permission to republish Integrating Newton's Equations in print or online must be granted by the author in writing.
|
|
|
|
