A New Method to Construct Finite Difference Operator with High Accuracy

We utilize Lagrange basis functions instead of Taylor expansion or optimization to obtain difference operator with high even accuracy. The general compact formulas that can be used to solve the first and second differential are presented owing to the good attribute of Lagrange basis. They also provide the option to calculate difference coefficient for arbitrary grids. As a special case, a uniform grid points are given, thus it generates fairly brief expressions of difference coefficient which display odd symmetry (first derivative) and even symmetry (second derivative), and these characters help to form a concise and straightforward differential scheme. The numerical results show that increasing order up to a certain level cannot improve the accuracy correspondingly but enlarge the computation due to the limit of coefficients at the far end approaches zero. So it is no need to apply higher order (e.g. 10 and more) for pursuing accuracy.