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High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems

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Abstract

We construct in this paper two efficient spectral algorithms in the frequency space for solving unsteady advection–reaction–diffusion equations with constant and variable coefficients. We first consider a Jacobi–Galerkin method for solving linear equations with constant coefficients. We then develop the direct solution algorithm for the linear advection–reaction–diffusion equations with variable coefficients using the Jacobi–Galerkin method with numerical integration. The proposed Jacobi–Galerkin methods, both in temporal and spatial discretizations, are successfully developed to handle the two-dimensional unsteady advection–reaction–diffusion equations with constant and variable coefficients and with fractional orders. In these methods, the model solution is expanded in both space and time in terms of polynomials bases built upon a linear combination of Jacobi polynomials. The homogeneous initial and Dirichlet boundary conditions are satisfied exactly by expanding the model solution in terms of these polynomials. The proposed Jacobi–Galerkin methods yield an exponential rate of convergence when the solution is smooth and allow a great flexibility to handle multi-dimensional time fractional advection–reaction–diffusion equations. Finally, a series of numerical examples are presented to demonstrate the efficiency and flexibility of the methods.

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Acknowledgements

We would like to thank anonymous referees for their careful reading of this paper and their many valuable comments and suggestions for improving the presentation of this work.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Matrouh University, Matrouh, Egypt

    Ramy M. Hafez

  2. Department of Applied Mathematics, National Research Centre, Dokki, Giza, 12622, Egypt

    Mahmoud A. Zaky

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  1. Ramy M. Hafez
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  2. Mahmoud A. Zaky
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Correspondence to Mahmoud A. Zaky.

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Hafez, R.M., Zaky, M.A. High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems. Engineering with Computers 36, 1813–1829 (2020). https://doi.org/10.1007/s00366-019-00797-y

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