### An improved numerical method for a 2D pollution water model: Direct model

#### Abstract

In this paper a 2D pollution water model with an improved numerical method is considered. In order to reduce the approximation errors of the numerical scheme, a new approximation method is introduced to calculate the concentration flux between two cells (*j*-cell and *l*-cell) in the direction of the normal vector \(\vec {n}\) orthogonal to their common side. The advantage of this approximation is that the concentration flux \({\partial C}/{\partial \vec {n}}\) from *j*-cell to *l*-cell and the other one from *l*-cell to *j*-cell are only different by their signs but not by their absolute values. Therefore, the errors of concentration simulated by this scheme are reduced and less than those obtained by a normal differential implicit discretization. This improvement of the scheme will be illustrated by two test cases. In the numerical tests we will display the difference between the exact solution, the classical scheme and the proposed scheme. Numerical results demonstrate the improvement of this approach.

#### Keywords

#### Full Text:

PDF#### References

D. Toro, C.Winfield, and R. B. Ambrose. QUAL2K. Brown and Barnwell,Water Quality Analysis Simulation Program, United States Environmental Protection Agency, (1987).

T. T. Ha and N. H. Phong. 2D-Model of contaminant water transmission processes and numerical simulation on a natural lake. Vietnam Journal of Mechanics, 32, (3), (2010), pp. 157–166. doi:10.15625/0866-7136/32/3/304.

T. H. Tran, D. T. Pham, H. V. Lai, and H. P. Nguyen.Water pollution estimation based on the 2D transport-diffusion model and the singular evolutive interpolated Kalman filter. Comptes Rendus Mécanique, 342, (2), (2014), pp. 106–124. doi:10.1016/j.crme.2013.10.007.

W. Rauch, M. Henze, L. Koncsos, P. Reichert, P. Shanahan, L. Somlyódy, and P. Vanrolleghem. River water quality modelling: I. State of the art.Water Science and Technology, 38, (11), (1998), pp. 237–244.

C. P. Dullemond and J. A. Hydrodynamics II: Numerical methods and applications. University of Heidelberg Summer Semester, (2007).

T. Stocker. Introduction to climate modeling. Springer-Verlag Berlin Heidelberg, (2011).

C. Licht, T. T. Ha, and Q. P. Vu. On some linearized problems of shallow water flows. Differential and Integral Equations, 22, (3/4), (2009), pp. 275–283.

P. A. Sleigh, P. H. Gaskell, M. Berzins, and N. G.Wright. An unstructured finite-volume algorithm for predicting flow in rivers and estuaries. Computers & Fluids, 27, (4), (1998), pp. 479–508. doi:10.1016/s0045-7930(97)00071-6.

E. F. Toro. Riemann problems and the WAF method for solving the two-dimensional shallow water equations. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 338, (1649), (1992), pp. 43–68. doi:10.1098/rsta.1992.0002.

X. Ying, J. Jorgeson, and S. S. Wang. Modeling dam-break flows using finite volume method on unstructured grid. Engineering Applications of Computational Fluid Mechanics, 3, (2), (2009), pp. 184–194. doi:10.1080/19942060.2009.11015264.

N. H. Phong, T. T. Ha, F. X. Le Dimet, and D. N. Hai. A wind-driven hydrodynamic and pollutant transport model with application of HLL Riemann solver schema. In International Conference on Engineering Mechanics and Automation (ICEMA 3), (2014), pp. 146–155.

D. Ambrosi. Approximation of shallow water equations by Roe’s Riemann solver. International Journal for Numerical Methods in Fluids, 20, (2), (1995), pp. 157–168. doi:10.1002/fld.1650200205.

G. Chen, H. Tang, and P. Zhang. Second-order accurate Godunov scheme for multicomponent flows on moving triangular meshes. Journal of Scientific Computing, 34, (1), (2008), pp. 64–86.

G. I. Marchuk and J. Ruzicka. Methods of numerical mathematics, Vol. 2. Springer-Verlag, New York, (1975).

C. Hirsch. Numerical computation of internal and external flows. Vol. 2: Computational methods for inviscid and viscous flows. JohnWiley & Sons, (1990).

B. Hunt. Dispersive sources in uniform ground-water flow. Journal of the Hydraulics Division, 104, (1978), pp. 75–85.

T. Karvonen. Pollutant transport in rivers. Department of Civil and Environmental Engineering, Helsinki University of Technology, (2002).

DOI: https://doi.org/10.15625/0866-7136/9714

### Refbacks

- There are currently no refbacks.

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam Tel: (+84) 24 3791 7103 Email: vjmech@vjs.ac.vn |