clipka <ano### [at] anonymousorg> wrote:=
> I guess the O(n^3) applies not to an increased "tank" size, but an
> increased resolution (which then also requires an increased resolution
> of time I guess), right?

Exactly.  It's the CFL (Courant-Friedrichs-Lewy) condition, and it basically
just says that information can't travel more than one grid cell per step.  This
means that if you double the resolution, you also have to cut the time step in
half to preserve stability.  So if you double the resolution, you have a factor
of four, and halving the time step makes 8.

> - So far I've used a more naive approach than what you propose,
> explicitly keeping track of both pressure and velocity (2 dimensions)
> separately (yet side-by-side in the same frame - hence the 3:1 aspect
> ratio), not realizing that the velocity can be inferred from the
> difference between previous frames (then again, can it? after all, the
> difference is an overlay of velocity influences in two dimensions...
> except of course if you process only a single dimension per step). This
> of course uses some weird hacks to properly superimpose the various
> dimensions.

Sorry for all the math.  I just popped my head in and saw something I couldn't
resist responding to.

Mathematically, you can rewrite the wave equation as two first order PDE's:

d^2p/dt^2 = d^2p/dx^2 + d^2p/dy^2 = del^2 p

becomes

dp/dt = div( v )
dv/dt = grad( p )

This is equivalent since differentiating the first, dp/dt, gives

d^2p/dt^2 = d/dt div(v) = div( dv/dt ) = div( grad(p) ) = del^2 p

Then, as you've done, you just store three quantities (i.e. pressure and 2 x
velocity) and compute them separately.  They're mathematically equivalent, but
computationally different.  For my approach, you need two previous frames but
only one quantity.  For your approach you need one previous frame, but three
quantities.

> Something along these lines, yes. Except that I'm not that deep into
> "official" math to know what a "laplacian" is

Sorry for the lingo; it's just the sum of second derivatives (curvature) in all
the different dimensions.  If the overall curvature is downward, it accelerates
downward, and vice versa.

 - Ricky

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