Numerical Solution of Advection-Dispersion Equation using Mesh-free Petrov-Galerkin Method (Case Study: Murray Burn River)

Abstract

Transport of pollutants in rivers is one of the most important issues in the environment. Many researchers have solved the advection-dispersion equation by various numerical methods, including finite difference and finite element methods. Despite their advantages, these methods also have disadvantages that are often related to netting of the problem domain. Therefore application of mesh-free methods which do not require solution domain network seems necessary. In the present study, the one-dimensional advection-dispersion equation has been solved using the Mesh-free Local Petrov-Galerkin method in the unsteady state. The Murray Burn River data were used to evaluate the performance of the model. The used approximation and weight function were the moving least squares function and the cubic spline function, respectively. In this study, 9 experiments were used to calibrate the model and 2 experiments were used to validate it. For calibration, the discharge coefficient and velocity plotted against the flow rate, and the power regression equation was extracted which had correlation coefficients of 0.925 and 0.988, respectively. In the validation mode, the dispersion coefficient and velocity were optimized by minimizing the mean squared error between computational and observational concentration for each model. The dispersion coefficient in this study was in the range of 0.13-1.1 m2/s for the flow rate of 13-437 L/s. The results indicated the acceptable performance and accuracy of mesh-free method.

Keywords

• cubic spline function
• mesh-free local petrov-galerkin method
• moving least squares function
• pollutant transport