The name is absent



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≡×fej

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.



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