Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation

In this paper simple polynomial interpolation is used to derive arbitrarily high-order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative (consisting of three second derivative nodes in the interior and two on the boundary) on non-uniform grids. Boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non-uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme. The high-order non-uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non-uniform grid with appropriate stretching so that more grid points are clustered near the boundary. The stability and resolution properties of the high-order non-uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented.