A spatial local method for solving 2D and 3D advection-diffusion equations

TL;DR

This paper introduces an implicit-explicit local differential transform method (IELDTM) that efficiently solves 2D and 3D advection-diffusion equations with high accuracy and fewer degrees of freedom, outperforming traditional methods.

Contribution

The study develops a novel IELDTM based on Taylor series and spectral collocation, optimizing local spatial degrees of freedom for improved accuracy in solving advection-diffusion equations.

Findings

IELDTM shows excellent convergence with h- and p-refinement.

Provides more accurate results with fewer degrees of freedom.

Outperforms finite difference, finite element, and spectral methods.

Abstract

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.