The name is absent

68

and

Z ≡ P^rU : RΛ → R^fe'

H ≡ U^τHU : R> → R^fe

R ≡ U^τW(P^τW)^¹ : R^fc' → R^fe∖

Since in (3.17) all are pointwise functions, the matrix P^τ 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,...,k_f

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

w^ω = [(2B(v°'-^υ) -Δt).≠-¹> + 2A(v°-^υ)Δt].∕[2B(v^c'-¹)) + Δt] (3.18)

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