A rational high-order compact ADI method for unsteady convection–diffusion equations Original Research Article

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) , the fourth-order ADI method of Karaa and Zhang (2004) and the fourth-order ADI method of Tian and Ge (2007) , but also proves more effective than the fourth-order Padé ADI method of You (2006) , in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.