Fourth Order Approximation of the Mixed Derivative Operator

We present a finite-difference approximation of the mixed derivative operator partial^2/partial x partial y. The scheme we use has the advantage over the Lagrange collocation scheme of reducing the number of arithmetic operations to the minimum necessary to accomplish the desired accuracy. One centred set of finite-differencing coefficients and one set which is unbalanced in both x and y are presented for the fourth order accurate differencing scheme. Test runs are performed which show the superior accuracy of the unbalanced scheme.