Numerical Simulation of Groundwater Pollution Problems Based on Convection Diffusion Equation

Lingyu Li, Zhe Yin
2017 American Journal of Computational Mathematics  
The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. To get the numerical solution, the Crank-Nicolson finite difference method is constructed, which is second-order accurate in time and space. Numerical simulation shows excellent agreement with the analytical solution. The dynamic visualization of the simulating results is realized on ArcGIS platform. This work provides a quick and intuitive
more » ... intuitive decision-making basis for water resources protection, especially in dealing with water pollution emergencies. consequently, the development of numerical solutions is required. S. Hasnain and M. Saqib originate results with finite difference schemes to approximate the solution of the classical Fisher Kolmogorov Petrovsky Piscounov (KPP) equation from population dynamics, their results show that Crank-Nicolson is very efficient and reliably numerical scheme for solving one-dimension fishers KPP equation [3]. During the last three decades, numerous transport problems have been solved numerical [4], theoretical numerical models are necessary tools were presented [5], the technique of intimating the movement of groundwater flow is improved greatly, see [6] [7] [8] and the references therein. Dillon gave many mathematical models and numerical methods for solving the groundwater problems [6]. Li and Jiao derived the analytical solutions of tidal groundwater flow in coastal two-aquifer system [7]. Sun applied a sort of numerical methods to simulate the movement of contaminants in groundwater [8]. At present, the researches on groundwater pollution problems are mainly divided into two categories at home and abroad [9] [10]. On one hand, people generally study various discrete numerical schemes of mathematical models (i.e. the corresponding partial differential equations) for groundwater pollution, by doing lots of related works, the numerical solution is obtained and the convergence is analyzed, Lin, L. et al. [9] derived a simplified numerical model of groundwater and solute transport. On the other hand, people develop the simulation software of groundwater numerical [11] [12] [13], particularly, in theory and application of important values, but also innovative, advanced and applied. In book [14] , Kovarik, K. sets his sights on reviewing the whole group of numerical methods from the oldest (the finite differences method), and discusses the basic equations of a groundwater flow and of the transport of pollutants in a porous medium. Therefore, we would like to use the finite difference method to study the numerical simulation methods about mathematical models with seepage of groundwater pollution. In order to improve the accuracy in the temporal direction, we propose a second-order scheme which based on centered Crank-Nicolson finite difference scheme [15]-[20]. And we simulate the water pollution problems in a certain area and verify the validity and practicability of the model and its algorithm. Meanwhile, the dynamic visualization of simulation results is realized on ArcGIS platform. We hope that our work can provide an important basis for water pollution accident emergency response and decision-making, and can be used for environmental protection personnel to deal with water pollution emergencies. The paper is organized as follows. In Section 2, we give the analytical solution of the Equations (1)-(5) by two-dimensional Fourier transform and the inverse Fourier transform. In Section 3, we introduce some notations, present the Crank-Nicolson finite difference discretization of the governing equation and derive the truncation error. Numerical experiment is given, exact solution comparisons with numerical solution are also discussed in Section 4. Visualization of simulation results based on GIS (ArcGIS figures) is presented in Section 5. Finally, conclusions and suggestions are drawn in Section 6.
doi:10.4236/ajcm.2017.73025 fatcat:46l7n4jtgrcyppvcpxnhjnb5ty