The name is absent

62

mV. Here A and B are pointwise functions whose actions are defined, recall (2.18), by

A(v)_i = uλ,_oo(v_i), B(v)_i =-r(v_i), z = l,..., N.              (3.4)

We discretize (3.2) (3.3) following the second-order staggered Euler scheme of
(Hines, 1984). More precisely, for a fixed time-step ∆i, we evaluate

ɑɑ) = G((J-3∕2)∆i), w^ɔ ≈ w((j-3∕2)∆f) and vɑɔ ≈ v((j-l)∆t), j = 1,2,...

via the marching scheme: Given wɑ^-1) and v^^-1∖ evaluate

wɑŋ = [(2B(√^j~¹>) - ∆i) .w(^j-ɪ) + 2A(√^j-¹))Δt]./ [2B(v°-ŋ) + ∆t]       (3.5)

and

v^ω = 2v_mj_d - v⁰~¹>                         (3.6)

where v_m∣_d is the solution to the linear system

(2∕∕∆i — H + diag(Φ(w^)e + G^))v_mj_d = 2v^^-1^∕∆t + Φ(w^)E_i + G^.E_s, (3.7)
where the ‘diag’ operator takes a vector and transforms it into a diagonal matrix. In

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