68
and
Z ≡ PrU : RΛ → Rfe'
H ≡ UτHU : R> → Rfe
R ≡ UτW(PτW)^1 : Rfc' → Rfe∖
Since in (3.17) all are pointwise functions, the matrix Pτ just picks off the entries at
the interpolation points z, and thus by recalling (3.4) we find
A(v)i ≡ w,,oo((Zv)i), B(v)i ≡ τ.((Zv)i), i = l,...,kf
and, similarly, Φ just computes the rows of Φ corresponding to the indices z. Hence
the reduced functions are of complexity kf, as desired.
We solve the reduced system using the same staggered Euler scheme. We denote
ɑθɔ = G((J-3∕2)∆t), ≈ w((J-3∕2)∆t) and vɑ'ɔ ≈ v((J-l)∆t), J = 1,2,...
and use the scheme: Given wʊ ŋ and vθ 1∖ evaluate
wω = [(2B(v°'-υ) -Δt).≠-1> + 2A(v°-υ)Δt].∕[2B(vc'-1)) + Δt] (3.18)
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