Stability and error estimates of a new high-order compact ADI method for the unsteady 3D convection–diffusion equation

Abstract

In this paper, a new high-order compact ADI method for the unsteady convection–diffusion equation in three dimension(3D) is considered. Collecting the truncation error of the finite difference operator by the recursion method, we derive a new high-order compact finite difference scheme for the unsteady 1D convection–diffusion equation firstly. Then, based on the ADI method, applying a correction technique to reduce the error of the splitting term, a high-order compact ADI method for the unsteady 3D convection–diffusion equation is proposed, in which we solve a series of 1D problems with strictly diagonal dominant tri-diagonal structures instead of the high-dimensional ones. The scheme is proved to be unconditional stable. Moreover, the regularity and error estimate of the numerical solution are derived. Finally, some numerical examples are performed, which confirm the theoretical prediction and show that much better computational accuracy results can be got by applying the new scheme.