Abstract
In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L<sub>1</sub>-norm) and L<sub>2</sub>-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
Highlights
Advection–diffusion equation is a parabolic partial differential equation which is derived on the principle of conservation of mass using Fick’s 1 law [11]
This paper aims to develop and present the accurate and stable numerical method which is Peaceman-Rachford alternating direct implicitly scheme that is capable of solving linear advection-diffusion equation and approximate the exact solution
Numerical results obtained by the present method has been computed with numerical results obtained by the methods in [1, 11, 15] and they are summarized and presented in Tables and graph
Summary
Advection–diffusion equation is a parabolic partial differential equation which is derived on the principle of conservation of mass using Fick’s 1 law [11]. Many harmful effects on humans and the environment such us atmospheric pollutions, contaminated flows in ground water aquifers, chemicals and migration of contamination in the sea waters and river systems, tracer dispersion in a porous medium are modeled by such types of equations [6, 11] It considers physical phenomena where in the diffusion process particles are moving with a certain velocity from higher concentration to lower concentration [11]. The treatments of this method present ever difficulties that have to be addressed to ensure the accuracy and stability of the solution To this end, this paper aims to develop and present the accurate and stable numerical method which is Peaceman-Rachford alternating direct implicitly scheme that is capable of solving linear advection-diffusion equation and approximate the exact solution. Section two is Statement of the problem, section three is Formulation of the numerical scheme, section four is Stability and convergence analysis for the approximate solution, section five is Numerical results, and section six is Discussion and conclusion
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