108
We take the SVD of this matrix, V = UΣXτ, and choose a strong reduced system
size ks < Ns. The POD matrix is then the first ks columns of U, i.e. U = (7(:, 1 : fcs).
Now we define the reduced variable by
vs = Uvs, vs ∈ Rfes, U ∈ R,jv≡×fe≈j
and then we substitute into (4.16) to obtain
∂t
Q I zs 0
(c∕2)C Hs
0
oʃl 1
Φ 2πα∆a?
ξ
⅞ιw(t)^
. ш.
0
N(Uvs(t), ws(t))
Multiplying through by the projection matrices U and Uτ, we obtain the IRKA+P0D
reduced system
where the new “Hines” matrix is now completely dense, but of dimension (kw + ks) ×
( kw 4- ks ).
∂t
(Zs 0) U ^
U7 HsU
’ 0_l , ɪ ∣^BIw(t)
UτΦ] 27rα∆rr [uτIs(t)
0
UrN(Uvs,ws)
(4.17)
The final step is to reduce the nonlinear term, which follows the same procedure
as in §3.2.2. In this way we obtain a basis for the nonlinear term by taking snapshots
of N, and then using the DEIM to obtain the set of ks spatial interpolation points z.