Rational High Order Compact Difference Method For Unsteady Convection Diffusion Equations

In this paper, a rational high-order compact (RHOC) difference method for solving convection diffusion equations is proposed. Based on Taylor series expansions for a function and the fourth order compact difference formula for the spatial discretization, we derivied an RHOC difference scheme for solving the one dimensional steady convection-diffusion equation. Then, by using the Crank-Nicolson scheme for the time discretization, an RHOC difference method is proposed for the one-dimensional unsteady convection-diffusion equation. The RHOC scheme is fourth-order accurate in space and second-order accurate in time. The von Neumann method is used to prove the RHOC scheme is unconditionally stable. Numerical experiments for some problems compared with previous other published schemes to demonstrate the accuracy and the stability of the present method. Then, based on the study of the one-dimensional unsteady convection-diffusion problem, we derived a rational high-order compact alternating direction implicit (RHOC-ADI) difference method for solving two-and three-dimensional unsteady convection diffusion equations. The proposed schemes are fourth-order accurate in space, second-order accurate in time and unconditionally stable. Numerical results show that the RHOC ADI difference method proposed for the unsteady convection-diffusion problems is not only can be applied to the unsteady convection diffusion problems, but also can be applied to solve pure diffusion or pure convection problems, and the computational results are superior to other difference schemes. The method combines well the advantages of high order compact scheme and ADI method, and provides an accurate, stable and efficient numerical method for solving unsteady convection diffusion equations. Finally, an RHOC difference scheme for solving the one-dimensional steady convection diffusion reaction equation is studied. By applying the Richardson extrapolation method and an operator interpolation technique, we improved the accuracy of the RHOC scheme from the fourth order to the sixth order. Numerical experiments are conducted to demonstrate the accuracy and the reliability of the present method.