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of each particle as being one molecule of the fluid. Nothing too special here!
Solids are almost always simulated in a Lagrangian way, with a discrete set
of particles usually connected up in a mesh.
The Eulerian approach, named after the Swiss mathematician Euler,
takes a different tactic that's usually used for fluids. Instead of tracking
each particle, we instead look at fixed points in space and see how mea-
surements of fluid quantities, such as density, velocity, temperature, etc.,
at those points change in time. The fluid is probably flowing past those
points, contributing one sort of change: for example, as a warm fluid moves
past followed by a cold fluid, the temperature at the fixed point in space
will decrease—even though the temperature of any individual particle in
the fluid is not changing! In addition the fluid variables can be changing
in the fluid, contributing the other sort of change that might be measured
at a fixed point: for example, the temperature measured at a fixed point
in space will decrease as the fluid everywhere cools off.
One way to think of the two viewpoints is in doing a weather report.
In the Lagrangian viewpoint you're in a balloon floating along with the
wind, measuring the pressure and temperature and humidity, etc., of the
air that's flowing alongside you. In the Eulerian viewpoint you're stuck on
the ground, measuring the pressure and temperature and humidity, etc., of
the air that's flowing past. Both measurements can create a graph of how
conditions are changing, but the graphs can be completely different as they
are measuring the rate of change in fundamentally different ways.
Numerically, the Lagrangian viewpoint corresponds to a particle sys-
tem, with or without a mesh connecting up the particles, and the Eulerian
viewpoint corresponds to using a fixed grid that doesn't change in space
even as the fluid flows through it.
It might seem the Eulerian approach is unnecessarily complicated: why
not just stick with Lagrangian particle systems? Indeed, there are schemes,
such as vortex methods (see, e.g., [Yaeger et al. 86, Gamito et al. 95, An-
gelidis and Neyret 05, Park and Kim 05]) and smoothed particle hydrody-
namics (SPH) (see, e.g., [Desbrun and Cani 96, Muller et al. 03, Premoze
et al. 03]) that do this. However, the rest of this topic will stick mostly to
the Eulerian approach for a few reasons:
•
It's easier to analytically work with the spatial derivatives like the
pressure gradient and viscosity term in the Eulerian viewpoint.
•
It's much easier to numerically approximate those spatial derivatives
onafixedEulerianmeshthanonacloudofarbitrarilymovingpar-
ticles.
