Comparıson of explicit and implicit temporal discretization methods in modelling groundwater radionuclide transport with the method of lines

Modelling of radionuclide transport in groundwater is important in determining the effects that may be caused by the leakage of spent fuel from nuclear power plants from geological repositories. In this thesis, finite difference-based method of lines (FD-MOL) was used to model radionuclide transport. In this context, the finite difference (FD) approach with different orders was applied to the spatial variable of the partial differential equation. The temporal variable was adaptively discretized with explicit and implicit modified midpoint methods, and the performance of the two approaches was compared in terms of accuracy, h-stability, p-stability, and computation time. Three problems were solved, including a one-dimensional uranium decay chain, and two-dimensional convection-diffusion-reaction (CDR) and tritium release from a nuclear reactor. The implicit method provided results with high accuracy even with large time steps, while the explicit approach required small steps to ensure numerical stability. For the uranium decay chain case, numerical concentrations of (_"92" ^"234" )"U" , (_"90" ^"230" )"Th" and (_"88" ^"226" )"Ra" were calculated with root mean square (RMS) errors of "9.45×" 〖"10" 〗^"-6" , "4.08×" 〖"10" 〗^"-5" and "7.93×" 〖"10" 〗^"-6" , respectively, with a time step of 1000 years for the implicit and 2 years for the explicit approaches for a total period of 50000 years, where the eighth order FD approach employed 400 nodes. Both temporal approaches showed a high h-convergence. When computation times were compared, the implicit approach yield solutions in a fast manner for sparse FD meshes, while the explicit method was faster for dense FD node distributions. The uranium decay problem was solved for different groundwater velocity values (V=50, 150 and 250 m⁄year) and results with high accuracy were obtained with the explicit method. The error values in the CDR problem decreased continuously and rapidly with the increase in the number and order of finite difference points. With n=625 and p=8, the maximum and RMS errors of FD-MOL were calculated as 1.422×10^(-17) and 7.435×10^(-18), respectively. Although the explicit and implicit approaches provided equivalent accuracy and stability properties in the CDR problem, there was a significant difference in computation time. In contrast to the one-dimensional uranium decay chain problem, the explicit approach had a great advantage in terms of computation time at high n_x values, while the implicit approach was more advantageous in coarse mesh structures. The implicit and explicit approaches gave RMS errors of "4.34×" 〖"10" 〗^"-11" with computation times of 19177 s and 3853 s, respectively, when a fourth-order FD with a "25×25" spatial node distribution was utilized. Implicit FD-MOL solution obtained with 100 FD nodes and p=1 had low accuracy due to the coarse mesh induced erroneous results that emanated near the source and diffused through the problem domain. On the contrary, with 3600 spatial points and p=4, the selection of the appropriate finite difference node number and order showed that the FD-MOL performed effectively for this problem as well. In terms of computation time, the explicit approach was found to be advantageous. When the effect of different velocity and diffusion constant on the solution of the tritium release problem was analyzed, it was observed that the tritium concentration approached zero at a shorter distance with decreasing velocity and the tritium concentration was higher over a wider range with increasing diffusion constant. Therefore, the method produced physically meaningful results in different conditions. The method of lines provided accurate and stable numerical results for all the problems. Comparing the explicit and implicit methods used in time discretization, it is observed that the explicit method was more advantageous and faster in terms of computation time, while the implicit method showed a better performance in terms of stability.