which ∀ γ ∈ [0 , 1] yields and However, the accuracy of this discretization is unknown and should be studied. Again, a Taylor expansion of φ$_{j}$$_{+}$$_{l}$ around φ$_{j}$ shows that the O ( a $^{2}$) corrections to L$_{(}$$_{n}$$_{)}$ [applied to h$_{j}$ in Eq. (10)] are of the form Thus, the O ( a $^{2}$) correction to L ( γ ) (2) is 1 12 1+7 γ 1+ γ ∂ 4 $_{x}$h . It attains its minimum value ( 1 $_{12}$∂ 4 $_{x}$h ) precisely for γ = 0, namely for L$_{(1)}$ . What is then the convenience of a more complex prescription for the Laplacian? A wise criterion for choosing b$_{1}$ and b$_{2}$ in L$_{(2)}$ is making the O ( a $^{2}$) corrections vanish. This yields the prescription b$_{1}$ = 16, b$_{2}$ = − 1, known to be accurate up to corrections of O ( a $^{4}$) [32]. Carrying out the procedure sketched in Sec. II A, we obtain The O ( a $^{2}$) corrections to Q$_{(}$$_{n}$$_{)}$ are which also vanishes for b$_{1}$ = 16, b$_{2}$ = − 1. Since this discretization scheme fulfills the consistency conditions, is accurate up to O ( a $^{4}$) corrections, and its prescription is not more complex than the ones studied before, it is 10 2 4! ∑ n l $_{=1}$l$^{4}$b$_{l}$ n l $_{=1}$l$^{2}$b$_{l}$ [ 3 ( ∂ 2 $_{x}$h ) 2 + 4 ( ∂ 3 $_{x}$h ) ( ∂$_{x}$h ) ] , L$_{(2)}$ ( h$_{j}$ ) ≡ 4 3 L$_{(1)}$ ( h$_{j}$ ) − 1 12 ( H j +2 j + H j − 2 j ) , (30) Q$_{(2)}$ ( h$_{j}$ ) ≡ 2 3 [ ( H j +1 j ) 2 + ( H j − 1 j ) 2 ] − 1 24 [ ( H j +2 j ) 2 + ( H j − 2 j ) 2 ] . (31) 2 4! ∑ n l $_{=1}$l$^{4}$b$_{l}$ n l $_{=1}$l$^{2}$b$_{l}$ ∂ 4 $_{x}$h. Q ( γ ) $_{(2)}$( h$_{j}$ ) ≡ 1 2(1 + γ ) { (1 − γ ) [ ( H j +1 j ) 2 + ( H j − 1 j ) 2 ] + γ 2 [ ( H j +2 j ) 2 + ( H j − 2 j ) 2 ] } . (29) L ( γ ) $_{(2)}$( h$_{j}$ ) ≡ 1 1 + γ [ (1 − γ ) ( H j +1 j + H j − 1 j ) + γ 2 ( H j +2 j + H j − 2 j ) ] = 1 2(1 + γ ) { 2 L$_{(1)}$ ( h$_{j}$ ) + γ [ L$_{(1)}$ ( h$_{j}$$_{+1}$ ) + L$_{(1)}$ ( h$_{j}$ − $_{1}$) ]} (28) B. A more accurate discretization scheme