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Leading-order error terms in mean values (1)

Weighted arithmetic mean: [¯ut,θ]n+θ=θun+1+(1θ)un=u(tn+θ)+Rn+θ,Rn+θ=12u(tn+θ)Δt2θ(1θ)+O(Δt3). Standard arithmetic mean: [¯ut]n=12(un12+un+12)=u(tn)+Rn,Rn=18u(tn)Δt2+1384u(tn)Δt4+O(Δt6).

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