content_image/341276.jpg content_image/341275.jpg content_image/341274.jpg content_image/341272.jpg content_image/341273.jpg content_image/341271.jpg
which ∀ γ ∈ [0 , 1] yields 

<img src='content_image/341276.jpg'>

and 

<img src='content_image/341275.jpg'>

However, the accuracy of this discretization is unknown and should be studied. 

## B. A more accurate discretization scheme 

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 

<img src='content_image/341274.jpg'>

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. 

<img src='content_image/341272.jpg'>

<img src='content_image/341273.jpg'>

<img src='content_image/341271.jpg'>

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 

