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multaneously).
9
These of course no longer fit in the same structure we
used to stored Laplacian matrices earlier; see Chapter 11 for a discussion
of general sparse matrices. However, the real new problem is that some
of the off-diagonal entries are now positive instead of all being negative.
This causes problems for the simple incomplete Cholesky preconditioners
we have developed, when Δ
t
gets too large.
Rasmussen et al. [Rasmussen et al. 04] suggest improving this by split-
ting up the problem between implicit and explicit parts, recovering the
decoupled linear solves with Laplacian-like matrices. In continuous vari-
ables this boils down to solving
u
n
)
T
.
ρ
∇·
u
V
+
Δ
t
ρ
=
u
n
+
Δ
t
u
V
η
∇
∇·
η
(
∇
(8.6)
We leave it to the reader to expand out the discretization in this case.
9
Rasmussen et al. [Rasmussen et al. 04] report getting an unsymmetric matrix for
this problem, probably due to using collocated velocities all at the grid cell centers.
To combine this with the MAC grid for pressure solves, they average the staggered
velocities to the grid cell centers for most stages of the time step, then average the grid
cell centers back to the staggered locations for the pressure solve. While this continual
averaging introduces unwanted numerical dissipation when simulating inviscid flow, for
highly viscous fluids it is fairly innocuous.
