THE SENSITIVITY AND ACCURACY OF 4TH-ORDER FINITE-DIFFERENCE SCHEMES ON NONUNIFORM GRIDS IN ONE-DIMENSION

We construct local fourth-order finite difference approximations of fi rst and second derivatives, on nonuniform grids, in one dimension. The approximations are required to satisfy symmetry relationships that co me from the analogous higher-dimensional fundamental operators: the di vergence, the gradient, and the Laplacian. For example, we require tha t the discrete divergence and gradient be negative adjoint of each oth er, DIV = -GRAD, and the discrete Laplacian is defined as LAP = DIVGR AD. The adjointness requirement on the divergence and gradient guarant ees that the Laplacian is a symmetric negative operator, The discrete approximations we derive are fourth-order on smooth grids, but the app roach can be extended to create approximations of arbitrarily high ord er. We analyze the loss of accuracy in the approximations when the gri d is not smooth and include a numerical example demonstrating the effe ctiveness of the higher order methods on nonuniform grids.