Fourth order compact FD methods for convection diffusion equations with variable coefficients

Fourth order finite difference methods combined with an integrating factor strategy for steady convection and diffusion partial differential equations with variable coefficients in both 2D and 3D are proposed using uniform Cartesian grids. An integrating factor strategy is applied to transform the convection and diffusion PDE to a self-adjoint form. Then, a fourth order finite difference method is obtained through a second order scheme followed by the Richardson extrapolation. Another approach is a direct fourth order compact finite difference scheme. The developed integrating factor strategy provides an efficient way for dealing with large convection coefficients. Several numerical examples are presented to demonstrate the convergence order and compare the two fourth order methods.