A compact method for second-order boundary value problems on nonuniform grids

A compact finite difference procedure is developed for the numerical solution of second-order two point boundary value problems with mixed boundary conditions. The procedure is applied to a series of linear and nonlinear problems on both uniform and nonuniform grids. In all cases the global estimates of the rate of convergence tend to four. In the last part of the paper, the procedure is extended to a nonlinear partial differential equation, Burgerʹs equation.