In the paper, a high-order alternating direction implicit (ADI) algorithm is presented to solve problems of unsteady convection and diffusion. The method is fourth- and second-order accurate in space and time, respectively. The resulting matrix at each ADI computation can be obtained by repeatedly solving a penta-diagonal system which produces a computationally cost-effective solver. We prove that the proposed scheme is mass-conserved and unconditionally stable by means of discrete Fourier analysis. Numerical experiments are performed to validate the mass conservation and illustrate that the proposed scheme is accurate and reliable for convection-dominated problems.

提出了一种高阶交替方向隐式（ADI）算法来解决非稳态对流扩散问题。该方法在空间和时间上分别是四阶和二阶精度。可以通过反复求解五对角线系统来获得每次ADI计算时得到的矩阵，该系统产生了具有计算成本效益的求解器。通过离散傅里叶分析，我们证明了该方案是质量守恒的，并且是无条件稳定的。进行了数值实验以验证质量守恒，并说明了该方案对于对流占优的问题是准确而可靠的。