Abstract
In the paper, a high-order alternating direction implicit (ADI) algorithm is presented to solve problems of unsteady convection and diffusion. The method is fourth- and second-order accurate in space and time, respectively. The resulting matrix at each ADI computation can be obtained by repeatedly solving a penta-diagonal system which produces a computationally cost-effective solver. We prove that the proposed scheme is mass-conserved and unconditionally stable by means of discrete Fourier analysis. Numerical experiments are performed to validate the mass conservation and illustrate that the proposed scheme is accurate and reliable for convection-dominated problems.
Highlights
At the present time, the population explosion, the high standard of living, and the industrial expansion have led to the rise of pollutants into the aquatic environment
To explore the efficiency of the proposed DHOC-alternating direction implicit (ADI) scheme, we provide the comparison of numerical results using the present DHOC-ADI scheme and the original High-order compact (HOC)-ADI scheme, which are given in Tables 4 and 5
The numerical solutions obtained by the presented DHOC-ADI and the RHOC-ADI schemes produce the solution that is in good agreement with the exact solution for all cases (see Figs. 5– 7(a), (d)), while small pulse distortion can be observed by the EHOC-ADI scheme (see Figs. 5–7(c))
Summary
The population explosion, the high standard of living, and the industrial expansion have led to the rise of pollutants into the aquatic environment. High-order compact (HOC) finite difference methods happen to be one of the most commonly used numerical methods to solve the unsteady convection–diffusion equation, that may extremely enhance the approximate precision (see, e.g., [6,7,8,9,10,11,12]). Tian and Ge presented [16] an exponential high-order compact alternating direction implicit (EHOCADI) method used to solve 2D unsteady convection–diffusion equations. The logical HOC scheme with the ADI (RHOC-ADI) method to solve unsteady convection–diffusion equations was investigated by Tian [17] On this account, its great effectiveness on solution accuracy and computational efficiency was demonstrated. According to the benefit of ADI methods and conservative discretization simultaneously, the improvement of the solution accuracy in the spatial domain based on the HOCADI method, which is a compact sixth-order scheme that inherits mass preserving properties, will be focused on. Remarkable conclusions are summarized to finalize the research paper
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