Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions
This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter ε. The approxima...
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| Format: | Article |
| Language: | English |
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ATNAA
2023-03-01
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| Series: | Advances in the Theory of Nonlinear Analysis and its Applications |
| Subjects: | |
| Online Access: | https://atnaea.org/index.php/journal/article/view/24/24 |
| _version_ | 1797248573248634880 |
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| author | Nouria ARAR Nouria ARAR Abdelhamid TALAAT |
| author_facet | Nouria ARAR Nouria ARAR Abdelhamid TALAAT |
| author_sort | Nouria ARAR |
| collection | DOAJ |
| description | This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter ε. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when ε tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.
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| first_indexed | 2024-04-24T20:16:44Z |
| format | Article |
| id | doaj.art-ce6eee11bd424f278440c7ed7c2ebd29 |
| institution | Directory Open Access Journal |
| issn | 2587-2648 |
| language | English |
| last_indexed | 2024-04-24T20:16:44Z |
| publishDate | 2023-03-01 |
| publisher | ATNAA |
| record_format | Article |
| series | Advances in the Theory of Nonlinear Analysis and its Applications |
| spelling | doaj.art-ce6eee11bd424f278440c7ed7c2ebd292024-03-22T16:51:25ZengATNAAAdvances in the Theory of Nonlinear Analysis and its Applications2587-26482023-03-017113314710.31197/atnaa.1139533Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditionsNouria ARAR0Nouria ARAR1Abdelhamid TALAAT2Laboratory of Applied Mathematics and History and Didactics of MathematicsLaboratory of Applied Mathematics and History and Didactics of MathematicsChinese Academy of Sciences, ChinaThis paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter ε. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when ε tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions. https://atnaea.org/index.php/journal/article/view/24/24spectral methodgalerkinrobin's conditionsadvection-diffusion equationgauss-quadraturedifference schemecrank nicolson scheme |
| spellingShingle | Nouria ARAR Nouria ARAR Abdelhamid TALAAT Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions Advances in the Theory of Nonlinear Analysis and its Applications spectral method galerkin robin's conditions advection-diffusion equation gauss-quadrature difference scheme crank nicolson scheme |
| title | Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions |
| title_full | Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions |
| title_fullStr | Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions |
| title_full_unstemmed | Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions |
| title_short | Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions |
| title_sort | efficient spectral legendre galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed robin boundary conditions |
| topic | spectral method galerkin robin's conditions advection-diffusion equation gauss-quadrature difference scheme crank nicolson scheme |
| url | https://atnaea.org/index.php/journal/article/view/24/24 |
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