Kamran^1,*, Farman Ali Shah¹, Kallekh Afef ², J. F. Gómez-Aguilar ³, Salma Aljawi⁴, Ioan-Lucian Popa^5,6,*

CMES-Computer Modeling in Engineering & Sciences, Vol.143, No.3, pp. 3433-3462, 2025, DOI:10.32604/cmes.2025.064815 - 30 June 2025

Abstract In this article, we develop the Laplace transform (LT) based Chebyshev spectral collocation method (CSCM) to approximate the time fractional advection-diffusion equation, incorporating the Atangana-Baleanu Caputo (ABC) derivative. The advection-diffusion equation, which governs the transport of mass, heat, or energy through combined advection and diffusion processes, is central to modeling physical systems with nonlocal behavior. Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain, eliminating the need for classical time-stepping methods that often suffer from stability constraints. For spatial discretization, we employ the CSCM, where More >

Xinyi Guan¹, Shaoqiang Tang^1,*

The International Conference on Computational & Experimental Engineering and Sciences, Vol.29, No.1, pp. 1-1, 2024, DOI:10.32604/icces.2024.010869

Abstract Inheriting a convergence difficulty explained by the Kolmogorov N-width [1], the advection-diffusion equation is not effectively solved by the Proper Generalized Decomposition [2] (PGD) method. In this paper, we propose a new strategy: Proper Generalized Decomposition with Coordinate Transformation (CT-PGD). Converting the mixed hyperbolic-parabolic equation to a parabolic one, it resumes the efficiency of convergence for advection-dominant problems. Combining PGD with CT-PGD, we solve advection-diffusion equation by much fewer degrees of freedom, hence improve the efficiency. The advection-dominant regime and diffusion-dominant regime are quantitatively classified by a threshold, computed numerically. Moreover, we find that appropriate More >

Guofei Pang¹, Wen Chen^1,2, K.Y. Sze³

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.4, pp. 299-322, 2014, DOI:10.3970/cmes.2014.097.299

Abstract Space-fractional advection-diffusion equation is a promising tool to describe the solute anomalous transport in underground water, and it has been extended to multi-dimensions with the help of weighted, fractional directional diffusion operator [Benson, Wheatcraft and Meerschaert (2000)]. Due to the nonlocal property of the space-fractional derivative, it is always a challenge to develop an efficient numerical solution method. The present paper extends the polynomialbased differential quadrature and cubature methods to the solution of steady-state spatial fractional advection-diffusion equations on a rectangular domain. An improved differential cubature method is proposed which accelerates the solution process considerably. More >