Resulting scheme: [Dtu]n−12=vn−12,[Dtv]n=1m(F(tn)−β|vn−12|vn+12−s(un)).
The truncation error in each equation is found from [Dtue]n−12=ve(tn−12)+Rn−12u,[Dtve]n=1m(F(tn)−β|ve(tn−12)|ve(tn+12)−s(un))+Rnv. Using (3)-(4) for derivatives and (21)-(22) for the geometric mean: ue′(tn−12)+124ue‴(tn−12)Δt2+O(Δt4)=ve(tn−12)+Rn−12u, and ve′(tn)=1m(F(tn)−β|ve(tn)|ve(tn)+O(Δt2)−s(un))+Rnv.