Robust Numerical Approaches for Time-Fractional Advection-dispersion Equations: Stability and Convergence of Implicit-explicit Schemes with Variable Coefficients

used in modelling subdiffusive phenomena taking place in heterogeneous media. The IMEX scheme is a very good compromise between computational performance and stability because we solve the stiff part of the problem implicitly and the other part, which is the advection term, explicitly. Using stringent stability criteria and based on a generalised Courant Friedrichs Lewy (CFL) condition, we find that the scheme is stable over an extended range of the time and spatial step sizes but becomes sensitive in the presence of variable coefficients. L2 and L∞ norms (and error norms) are used to verify convergence where first-order accuracy in time and second-order accuracy in space are achieved as expected theoretically. Numerical analysis shows the aptitude of a scheme to portray the sub-diffusive nature specific to fractional dynamics, despite the change in the fractional order. It can be observed that comparative analysis demonstrates the strength of the IMEX scheme under constant and variable coefficient conditions, and results suggest that, although the variable coefficient case results in minor increases in error and computation time, the scheme remains stabilised and efficient. The plots of stability regions, error surfaces, and solution dynamics give necessary information about the effect of uncertainty in the coefficient and choice of fractional order on numerical behaviour. It is the work that triggers the IMEX finite difference scheme as the sound method of simulating complicated subdiffusive processes, which can be applied to diverse areas (ground waves and pollutants transport). A possible line of future research can expand the utility of the framework to multi-dimensional domains and adaptive meshing approaches.