Geometric mean: [¯u2t,g]n=un−12un+12=(un)2+Rn,Rn=−14u′(tn)2Δt2+14u(tn)u″(tn)Δt2+O(Δt4).
Harmonic mean: [¯ut,h]n=un=21un−12+1un+12+Rn+12,Rn=−u′(tn)24u(tn)Δt2+18u″(tn)Δt2.