Graphics Reference
In-Depth Information
This is precisely the motivation for splitting: we may not be able to easily
deal with the complexity of the whole equation, but it's built out of separate
terms that we
do
have good methods for. I'll call the special integration
algorithms
F
(Δ
t, r
)and
G
(Δ
t, s
). Our splitting method is then
q
=
F
(Δ
t, q
n
)
,
(2.6)
q
n
+1
=
G
(Δ
t, q
)
.
(2.7)
If
F
() and
G
() were just forward Euler, then this is exactly the same as
Equations (2.4) and (2.5), but the idea is again that they're something
better. If you do the Taylor series analysis, you can show we still have a
first-order-accurate method
2
but I'll leave that as an exercise.
Splitting really is just the principle of divide-and-conquer applied to
differential equations: solving the whole problem may be too hard, but
you can split it into pieces that are easier to solve and then combine the
solutions.
If you're on the ball, you might have thought of a different way of com-
bining the separate parts: instead of sequentially taking the solution from
F
() and then plugging it into
G
(), you could run
F
() and
G
() in parallel and
add their contributions together. The reason we're
not
going to do this
but will stick to sequentially working through the steps, is that our spe-
cial algorithms (the integrators
F
() and
G
() in this example) will guarantee
special things about their output that are needed as preconditions for the
input of other algorithms. Doing it in the
right
sequence will make every-
thing work, but doing it in parallel will mess up those guarantees. We'll
talk more about what those guarantees and preconditions are in the next
section.
2.2 Splitting the Fluid Equations
We're going to use splitting on the incompressible fluid equations. In par-
ticular, we'll separate out the advection part, the body forces (gravity)
part, and the pressure/incompressibility part.
If viscosity is important,
we'll also separate it out: see Chapter 8.
2
There are more complicated ways of doing splitting in fluid dynamics, which can get
higher-order accuracy, but for now we won't bother with them. At the time of writing
this topic, this has remained a sadly overlooked area for improvement within graphics:
the first-order time-splitting error can be very significant indeed.
