## Abstract

The spectral Legendre–Galerkin method for solving a two-dimensional nonlinear system of advection–diffusion–reaction equations on a rectangular domain is presented and compared with analytical solution. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and computation of the nonlinear terms in the Chebyshev–Gauss–Lobatto points. The main difference of the spectral Legendre–Galerkin method presented in the current paper with the classic Legendre–Galerkin method is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the spectral Legendre–Galerkin method the nonlinear terms are efficiently handled using the Chebyshev–Gauss–Lobatto points and also the boundary conditions are imposed strongly as collocation methods. Combination of the proposed method with a semi-implicit time integration method such as the Leapfrog–Crank–Nicolson scheme leads to reducing the complexity of computations and obtaining a linear algebraic system of equations. Efficiency and spectral accuracy of the proposed method are demonstrated numerically by some examples.

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The author is grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.

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Fakhar-Izadi, F. An efficient spectral-Galerkin method for solving two-dimensional nonlinear system of advection–diffusion–reaction equations.
*Engineering with Computers* **37**, 975–990 (2021). https://doi.org/10.1007/s00366-019-00867-1

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DOI: https://doi.org/10.1007/s00366-019-00867-1