A unified framework to generate optimized compact finite difference schemes

A unified framework to derive optimized compact schemes for a uniform grid is presented. The optimal scheme coefficients are determined analytically by solving an optimization problem to minimize the spectral error subject to equality constraints that ensure specified order of accuracy. A rigorous stability analysis for the optimized schemes is also presented. We analytically prove the relation between order of a derivative and symmetry or skew-symmetry of the optimal coefficients approximating it. We also show that other types of schemes e.g., spatially explicit, and biased finite differences, can be generated as special cases of the framework.