Compact finite-difference scheme for differential relations' approximation

Differential relations include both differential operators and solvers of boundary value problems. The formulas of compact finite-difference approximations for differential relations of the first and second orders are obtained. Three-point stencils are used. Like classical finite difference schemes, the tridiagonal matrix is inverted to implement the scheme. However, compact schemes provide significantly higher accuracy and order of the 4th approximation instead of the 2nd.