A New High-order Compact ADI Method For The Unsteady 3D Convection-diffusion Equation

Convection-diffusion phenomenon has always been the focus of attention in the field of the fluid mechanics.However,due to the constraints of many practical physical condition,the convection-diffusion problem tends to have convection-dominated property,which leads to the non-physical oscillation when using the classical schemes.A new kinds of finite difference methods(abbreviated as NM methods)called the new highorder compact alternating direction implicit(ADI)for this problem will be considered in this paper.Firstly,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.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 stability 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.