A Family of High Order Finite Difference Schemes with Good Spectral Resolution

This paper presents a family of finite difference schemes for the first and second derivatives of smooth functions. The schemes are Hermitian and symmetric and may be considered a more general version of the standard compact (Padé) schemes discussed by Lele. They are different from the standard Padé schemes, in that the first and second derivatives are evaluated simultaneously. For the same stencil width, the proposed schemes are two orders higher in accuracy, and have significantly better spectral representation. Eigenvalue analysis, and numerical solutions of the one-dimensional advection equation are used to demonstrate the numerical stability of the schemes. The computational cost of computing both derivatives is assessed and shown to be essentially the same as the standard Padé schemes. The proposed schemes appear to be attractive alternatives to the standard Padé schemes for computations of the Navier–Stokes equations.