Flows of molecular fluids in nano- structured channels

X = x i
Xm = x (m) i (Δt)m /m! The m’th derivative m=0,1,2,3,4,5
X P (t+Δt)=X(t)+X1(t)+X2(t)+X3(t)+X4(t)+X5(t)
X1 P (t+Δt)=X1(t)+2X2(t)+3X3(t)+4X4(t)+5X5(t)
X2 P (t+Δt)=X2(t)+3X3(t)+6X4(t)+10X5(t)
X3 P (t+Δt)=X3(t)+4X4(t)+10X5(t)
X4 P (t+Δt)=X4(t)+5X5(t)
X5 P (t+Δt)=X5(t)
Error: ΔX2=X2(t+Δt)-X2(t+Δt)
X(t+Δt)= X P (t+Δt) + α 0
ΔX2
X1(t+Δt)=X1 P (t+Δt) + α 1
ΔX2
X2(t+Δt)=X2 P (t+Δt) + α 2
ΔX2
X3(t+Δt)=X3 P (t+Δt) + α 3
ΔX2
X4(t+Δt)=X4 P (t+Δt) + α 4
ΔX2
X5(t+Δt)=X5 P (t+Δt) + α 5
ΔX2
α m
depend on the degree of the differential
equations and on the order of the Taylor
series used: to ensure numerical stability
5’th order:
3/16, 251/360, 1, 11/18, 1/6, 1/60
3’rd order:
1/6, 5/6, 1, 1/3