Numerical Simulation for Nonlinear Space-Fractional Reaction Convection-Diffusion Equation with Its Application

In this research, we adopt a fully implicit approach with a weighted shifted Grunwald–Letnikov difference operator to find the numerical solution of the one and two-dimensional nonlinear space fractional convection–diffusion-reaction equation over a bounded domain. We employ the Peacemann-Rachford alternative direction implicit technique for 2D problems, which is computationally efficient and is used to reduce a 2D problem to a 1D problem by alternately applying x and y directions. This type of problem allows us to successfully predict and investigate pollutant distribution in groundwater, which is based on two processes: convection and diffusion. Because computing the numerical solution of the nonlinear system of equations at each time step using the fixed-point iteration technique is computationally expensive, we have linearized our problem. The numerical scheme’s unconditional stability and second-order convergence in both space and time are theoretically investigated. Finally, numerical experiments with several variations of the nonlinear reaction term are performed, confirming the theoretical analysis and demonstrating the effectiveness of the suggested strategy.