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 scheme coefficients are determined analytically by solving an optimization problem minimize the spectral error subject equality constraints that ensure specified order of accuracy. rigorous stability analysis also We show other types e.g., spatially explicit and biased finite differences, can be generated as special cases framework. Optimized using this tested on canonical partial differential equations, numerical results compared with standard schemes.