A finite volume–complete flux scheme for the time fractional advection–diffusion–reaction equation

For the numerical solution of the time-fractional advection–diffusion–reaction equation (TFADRE), we propose an implicit finite-volume complete flux scheme (FV-CFS). In our proposed scheme, the spatial discretization is done by applying the finite volume method, where the fluxes through the interfaces of the control volumes have been evaluated by solving suitable local inhomogeneous boundary value problems (BVPs), whereas a finite difference approximation of the time-fractional derivative, based on piecewise linear interpolation, is used for the temporal discretization. As the fluxes are evaluated by solving suitable BVPs, these fluxes come out as the sum of a homogeneous and an inhomogeneous part. Here, the homogeneous part represents the contribution of the advection and diffusion terms, whereas the inhomogeneous part represents the influence of various sources present in the governing equation. Analysis of consistency, stability, and a priori error estimates shows that the proposed scheme is conditionally stable, with a space–time order of convergence of O(Δt2−γ+Δx2), where Δx and Δt denote the spatial grid size and time step size, respectively. The scheme is validated with known solutions for various orders γ∈(0,1) of the time-fractional derivative, and the space–time order of accuracy has been realized through numerical simulations for benchmark problems with known analytical solutions in both one and two dimensions. In addition to the comparison with the pre-existing schemes, the space–time numerical capability is assessed for extreme values of important parameters like the Péclet number and orders of time-fractional derivatives γ with the appropriate justification of their physical significance. While demonstrating the success of simulated results for several examples, our scheme opens up possibilities for broader applications.