A Bound-Preserving Numerical Scheme for Space-Time Fractional Advection Equations

We develop and analyze an explicit finite difference scheme that satisfies a bound-preserving principle for space-time fractional advection equations with the orders of 0<alpha and beta <= 1. The stability (and convergence) of the method is discussed. Due to the nonlocal property of the fractional operators, the numerical method generates dense coefficient matrices with complex structures. In order to increase the effectiveness of the method, we use Toeplitz-like structures in the full coefficient matrix in a sparse form to reduce the costs of computation, and we also apply a fast evaluation method for the time-fractional derivative. Therefore, an efficient solver is constructed. Numerical experiments are provided for the utility of the method.