High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.