Numerical Modeling-Based Comparison of Leap-Frog and Implicit Crank-Nicolson Schemes for Instantaneous Spill of Pollutant in Rivers

Numerical modeling of pollutant spills that are released instantaneously in rivers has been commonly applied for water quality purposes. Recently, different numerical schemes have been used to solve for the river hydrodynamics from the shallow water equations (SWEs), affecting the longitudinal pollutant concentrations prediction of the advection-dispersion equation (ADE). In this study, two numerical schemes for solving the SWEs, Explicit Leap-Frog Scheme (ELFs) and Implicit Crank-Nicolson Scheme (ICNs), were implemented based on a field case study, and the pollutant concentrations distribution along the river were explored and compared to the ADE analytical solution. Results showed that the maximum concentration predicted by the ICNs decreased from 0.1071 to 0.0084 ppm after 5 and 8 days from the spill date, respectively, with an average flow velocity of 0.1545 m/sec. On the other hand, the maximum concentration predicted by the ELFs decreased from 0.1068 to 0.0083 ppm during the same period with an average flow velocity of 0.1550 m/sec. Accordingly. both schemes revealed good agreement compared to the analytical solution, for instance, at the simulation time of 8 days the ICNs statistical errors were RMSE of 0.000174 ppm and MAE of 0.0000771 ppm, while the ELFs errors were RMSE of 0.000182 ppm and MAE of 0.0000824 ppm. In terms of computational time, the ICNs spent higher cost of about 0.5982 sec during the simulation period of 8 days, while the ELFs took about 0.1738 sec for the same period. Furthermore, for both schemes as the longitudinal increments value of the finite difference grid increases, the model time step increases and the execution time decreases. Thus, it is necessary to choose time step and spatial increment length that obey the governing equation stability condition in order to conserve the concentrations distribution along the river spatially and temporally and make good predictions.