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In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (<italic>θ</italic>-method, <italic>θ</italic> ∈ [1/2, 1] (<italic>θ</italic> = 1 corresponds to the backward Euler method and <italic>θ</italic> = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders <italic>O</italic>(Δ<italic>t</italic> + <italic>h</italic>3) for the backward Euler method and <italic>O</italic>(Δ<italic>t</italic>2 + <italic>h</italic>3) for the Crank-Nicolson method, where Δ<italic>t</italic> and <italic>h</italic> are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.
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In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the backward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders O(Δt + h3) for the backward Euler method and O(Δt2 + h3) for the Crank-Nicolson method, where Δt and h are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.

A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problems

LESJEP Laboratory, FSJESJ, University Chouaïb Doukkali

ANAA Research Team, ESTO, LANO Laboratory, FSO, University Mohammed Premier

LAMSC Laboratory, FST, University Sultan Moulay Slimane

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The...

A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problems

Published Online: Jan 01, 2022

Page range: 15 - 34

Received: Mar 03, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0029

Keywords
convection-diffusion equation, -method, quintic, spline collocation method, convergence

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problems

Published Online: Jan 01, 2022

Page range: 15 - 34

Received: Mar 03, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0029

Keywordsconvection-diffusion equation, -method, quintic, spline collocation method, convergence

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
convection-diffusion equation, -method, quintic, spline collocation method, convergence