A numerical method for two-dimensional transient nonlinear convection-diffusion equations

In this research, we have developed the half-boundary method (HBM) for nonlinear convection-diffusion equations (CDEs), which hold significant importance in nuclear power engineering. The HBM employs a variable relationship between the nodes within the computational domain and the nodes located on half of the boundaries. This approach offers notable benefits, including the reduction of the maximum matrix order and the optimization the maximum memory storage for calculations. Moreover, the HBM is an efficient and streamlined approach to directly handle Neumann boundary conditions, thanks to the utilization of mixed variables. We primarily investigate the memory usage and accuracy of the proposed algorithm in the unsteady-state CDEs, in context of material nonlinear CDEs, the Burgers' equation and the system of Burgers' equations. The numerical results obtained demonstrate the method's potential in simulating flow and heat transfer phenomena, particularly in situations where convection is dominant.