The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension

Abstract

We construct local fourth-order finite difference approximations of first and second derivatives, on nonuniform grids, in one dimension. The approximations are required to satisfy symmetry relationships that come from the analogous higher-dimensional fundamental operators: the divergence, the gradient, and the Laplacian. For example, we require that the discrete divergence and gradient be negative adjoint of each other, DIV∗ = − GRAD , and the discrete Laplacian is defined as LAP = DIVGRAD . The adjointness requirement on the divergence and gradient guarantees that the Laplacian is a symmetric negative operator. The discrete approximations we derive are fourth-order on smooth grids, but the approach can be extended to create approximations of arbitrarily high order. We analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids.