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

Huseyin Tunc, Murat Sari*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


Purpose: This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations. Design/methodology/approach: The parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense. Findings: The IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods. Originality/value: This study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

Original languageEnglish
Pages (from-to)2068-2089
Number of pages22
JournalEngineering Computations
Issue number9-10
Publication statusPublished - 5 Dec 2023
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2023, Emerald Publishing Limited.


The first author would like to thank the Science Fellowships and Grant Programs Department of TUBITAK (TUBITAK BIDEB) for their support to the author's academic research. The second author thanks Dr Aniela Balacescu (Constantin Brancuşi University) for the hospitality.

FundersFunder number
Programs Department of TUBITAK


    • Advection-diffusion equation
    • Chebyshev collocation method
    • Hp-refinement
    • Multidimensional PDEs
    • Taylor series


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