An approach of Finding Second Derivative Finite Differences Compact Schemes

In solving problems from Computational Fluid Dynamics (CFD) and physics, huge efforts should be afforded to obtain accurate and applicable schemes for the derivatives. Based on the idea of high order polynomials, many sets of second derivative schemes are derived in this paper. These sets are grouped according to the order of the accuracy of the approximations from order three to order seven. Different types of second derivative forward, central, and backward compact with some traditinal approximations are inrtoduced at each set by the proposed method. The order of accuracy is verified of each scheme using the technique of finding the values of the coefficients for the error terms by matching both sides of the given scheme. Many schemes that are introduced in this article are applicable to some problems from science and engineering.