Abstract
In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.
Highlights
Consider the following advection-diffusion-reaction equation with given initial and boundary conditions:ܥ௧ ൌ ܥܦ௫௫ െ ܸܥ௫ െ ߠܥ, ݐ 0, ܽ ݔ ܾ (1)ܥሺݔ, 0ሻ ൌ ݂ሺݔሻ (2) ܥሺܽ, ݐሻ ൌ ݃ሺݐሻ or డ డ௫ ݄ሺݐሻ at ݔൌܽ
(1) is the Taylor-Galerkin method being effective for many problems represented by differential equations
After performing the time discretization, the Galerkin method is used for the spatial approximation by utilizing B-splines basis functions (5)
Summary
Consider the following advection-diffusion-reaction equation with given initial and boundary conditions:. When the advection is dominant to the diffusion, the exact solutions mostly fail and diverge. In these cases, the effective numerical methods need to be constructed to obtain accurate and stable results of the model equation. Various versions of finite element methods have profoundly been analyzed in the literature. In addition to finite element-based methods, some other numerical methods were taken into consideration in dealing with the ADR processes [1,10]. This study discovers some finite element based hybrid techniques to analyze the model problems encountered in broad range of science. All produced results are compared with the literature and exact solutions. Numerical methods continuity of the approximate solution and the first and second order-derivatives at all region
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