In this study, an implicit-explicit local differential transform method (IELDTM) based on Taylor series representations is produced for solving 2D and 3D advection-diffusion equations. The parabolic advection-diffusion equations are reduced to the nonhomogeneous elliptic system of partial differential equations with the utilization of the Chebyshev spectral collocation approach in temporal variable. The IELDTM is constructed over 2D and 3D meshes using continuity equations of the neighbour representations with either explicit or implicit forms in related directions. The IELDTM is proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over existing numerical techniques is the optimization of the local spatial degrees of freedom. It has been proven that IELDTM provides more accurate results with far less degrees of freedom than the finite difference, finite element and spectral methods.

翻译：在这项研究中,根据Taylor系列表示法,制作了一种基于Taylor系列表示法的隐含的当地差分变换法(IELDTM),用于解决2D和3D对流-扩散方程式;抛物线对流-扩散方程式,在时间变量中,利用Chebyshev光谱共移法,减为部分异差方程式的不对等离异的椭圆系统;IELDTM用邻方表示法的连续性方程式构建于2D和3D模件;IELDTM通过实验性地说明精度和精度结果,证明具有极好的趋同性;IELDTM相对于现有数字技术的一个显著特点是优化当地空间自由度;已经证明,IELDTM所提供的自由度远低于有限差异、有限元素和光谱方法。