An efficient two-level factored method for advection-dispersion problem with spatio-temporal coefficients and source terms

Eric Ngondiep; Eric Ngondiep

doi:10.3934/math.2023582

AIMS Mathematics

2023, Volume 8, Issue 5: 11498-11520. doi: 10.3934/math.2023582

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An efficient two-level factored method for advection-dispersion problem with spatio-temporal coefficients and source terms

Eric Ngondiep ^{1,2
,
,}

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), 90950 Riyadh 11632, Saudi Arabia
2.
Hydrological Research Centre, Institute for Geological and Mining Research, Yaounde 4110, Cameroon

Received: 09 September 2022 Revised: 10 February 2023 Accepted: 24 February 2023 Published: 14 March 2023
MSC : 35K20, 65M06, 65M12

A two-level factored implicit scheme is considered for solving a two-dimensional unsteady advection-dispersion equation with spatio-temporal coefficients and source terms subjected to suitable initial and boundary conditions. The approach reduces multi-dimensional problems into pieces of one-dimensional subproblems and then solves tridiagonal systems of linear equations. The computational cost of the algorithm becomes cheaper and makes the method more attractive. Furthermore, the two-level approach is unconditionally stable, temporal second-order accurate and spatial fourth-order convergent. The developed numerical scheme is faster and more efficient than a broad range of methods widely studied in the literature for the considered initial-boundary value problem. The stability of the proposed procedure is analyzed in the $ L^{\infty}(t_{0}, T_{f}; L^{2}) $-norm whereas the convergence rate of the algorithm is numerically analyzed using the $ L^{2}(t_{0}, T_{f}; L^{2}) $-norm. Numerical examples are provided to verify the theoretical result.
Citation: Eric Ngondiep. An efficient two-level factored method for advection-dispersion problem with spatio-temporal coefficients and source terms[J]. AIMS Mathematics, 2023, 8(5): 11498-11520. doi: 10.3934/math.2023582

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Abstract

A two-level factored implicit scheme is considered for solving a two-dimensional unsteady advection-dispersion equation with spatio-temporal coefficients and source terms subjected to suitable initial and boundary conditions. The approach reduces multi-dimensional problems into pieces of one-dimensional subproblems and then solves tridiagonal systems of linear equations. The computational cost of the algorithm becomes cheaper and makes the method more attractive. Furthermore, the two-level approach is unconditionally stable, temporal second-order accurate and spatial fourth-order convergent. The developed numerical scheme is faster and more efficient than a broad range of methods widely studied in the literature for the considered initial-boundary value problem. The stability of the proposed procedure is analyzed in the $ L^{\infty}(t_{0}, T_{f}; L^{2}) $-norm whereas the convergence rate of the algorithm is numerically analyzed using the $ L^{2}(t_{0}, T_{f}; L^{2}) $-norm. Numerical examples are provided to verify the theoretical result.

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