An interpolation based finite difference method on non-uniform grid for solving Navier–Stokes equations

• A general method to derive combined compact difference schemes on non-uniform grid.
• Scheme covers arbitrary stencil and achieves arbitrary accuracy order.
• New non-uniform grid with both clustering for stability and stretching in far field.
• Simulates boundary layer transition problems much more efficiently.

This paper presents a Hermite polynomial interpolation based method to construct high-order accuracy finite difference schemes on non-uniform grid. This method can achieve arbitrary order accuracy by expanding the grid stencil and involving higher order derivatives. The paper first constructs combined compact difference schemes, from which compact difference schemes and super-compact difference schemes are shown to be derived by linear operations. Explicit schemes are further shown to be particular cases of this interpolation method. Using the present derivation method, previously reported classical schemes can be constructed on non-uniform grid and a new 5-point combined compact difference scheme is developed in particular. A new 2-piecewise function is also provided for non-uniform grid generation. The first piece of function stabilizes the scheme on Dirichlet boundary by clustering the grid points appropriately and the second piece is to stretch the outer grids according to the simulation domain of interest. This new scheme with non-uniform grid shows excellent stability properties and high spectral resolution as compared with other classical compact and combined compact difference schemes. To further demonstrate the present scheme, simulation of boundary layer transition problems using the three-dimensional incompressible Navier–Stokes equations is performed and good agreement with experimental results is obtained.