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A finite difference formula for the discretization of d^3/dx^3 on nonuniform grids

We analyze the use of a five-point difference formula for the dis- cretization of the third derivative operator on nonuniform grids. The formula was derived so as to coincide with the standard five-point formula on regular grids and to lead to skew-symmetric schemes. It is shown that, under periodic boundary conditions, the formula is supraconvergent in the sense that, in spite of being inconsistent, it gives rise to schemes with second-order convergence. However, such a supraconvergence only takes place when the number of points in the grid is odd: for grids with an even number of points the inconsistency of the formula results in lack of convergence. Both stationary and evolutionary problems are considered and the analysis is backed by numerical experiments.