Non-standard schemes for time-fractional reaction–advection–diffusion problems

The present paper concerns the numerical solution of time-fractional reaction–advection–diffusion problems. In real applications, an important issue is the preservation of qualitative properties of the analytical solution, such as positivity, which standard methods achieve only for small stepsizes. Here, novel explicit and implicit non-standard finite difference methods are introduced, by treating different terms in the approximations on different time levels, in a way to keep the solution non-negative at all times. A rigorous analysis of the stability and convergence of the proposed schemes is provided, offering robust theoretical results that illustrate their effectiveness in preserving positivity while generating accurate approximations of the solution. Finally, some numerical experiments demonstrate the efficacy of the proposed methods on different benchmark problems.