Consider the following, very generic pair of differential equations that describe the dynamical motion of a particle (or a group of particles) in the presence of some "external" force F, which almost always is a function of the particle's position and sometimes also depends on the velocity of the particle.
Dtx = v
It is this pair of equations that must be integrated forward in time in a self-consistent fashion when you're performing classical "N-body" (e.g., molecular dynamics or stellar dynamics) simulations. Rather than modeling a system that is composed of discrete particles (which is usually covered in the PHYS7411), here we will consider how to follow the time-evolution of a continuum fluid that moves under the influence of an external force.
First, let's discuss the difference between "Lagrangian" and "Eulerian" representations of the equation of motion. Then let's walk through a sampling of the wide variety of ways the equation of motion can be written before making a tactical decision regarding the form of the equation that we will choose to write in finite-difference form.
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