Ahigh-order Compact ADI Method On Non-uniform Grid For Unsteady Convection-diffusion Equations

In this paper, a high-order compact ADI difference method on non-uniform grid is studied for solving the unsteady convection-diffusion equations. The method combines well the advantage of high-order compact scheme and ADI method, and provides an accurate and efficient numerical method for solving the unsteady convection-diffusion equations. Firstly, on the basis of Taylor series for function and method of reminder-correction, we derive the high order compact scheme on non-uniform grid for the one-dimensional (1D) steady convection diffusion equation. Then the second-order backward Eulerian difference method is employed for the discretization of temporal derivatives, we propose a high order compact scheme on non-uniform grid for the1D unsteady convection diffusion equation. Based on the work for the1D problem, we extend it to the2D and3D unsteady convection diffusion equations and derive a high order compact ADI scheme on non-uniform grid by using Crank-Nicolson method for the time discretization. The present ADI scheme is the second-order in time and the third to fourth-order in space. The ADI scheme corresponds to a series of tridiagonal linear equations which can be inverted by multiple use of the chasing method. The numerical experiments show that there is a good character of adaptability to boundary layer and large gradient problems of the proposed scheme. A higher accuracy can be obtained by choosing the right mesh generation function and adjusting the grid distribution parameters reasonably. The employment of ADI method for high dimensional problems results in a very efficient solver with a considerable saving in computing time.