A Positivity-Preserving and Robust Fast Solver for Time-Fractional Convection–Diffusion Problems

This paper presents a fast solver for time-fractional two-dimensional convection-diffusion problems that maintains non-negativity of numerical solutions. To this end, two new techniques are developed. (i) A three-part decomposition of the L1 discretization for Caputo derivatives is proposed and justified for fast evaluation while maintaining positivity; (ii) A positivity-correction technique is devised for both diffusive and convective fluxes. An upwinding technique for the bilinear finite volume approximation on general quadrilaterals is utilized for enabling the solver robustness in handling convection dominance. The solver attains optimal convergence rates when graded temporal meshes are used. These properties are theoretically justified and numerically illustrated.