Temporal Second-Order Fast Finite Difference/Compact Difference Schemes for Time-Fractional Generalized Burgers' Equations

Two kinds of numerical schemes are investigated for time-fractional generalized Burgers' equations (TFGBE). The first kind is obtained by the temporal second-order fast finite difference approach for the TFGBE with Dirichlet boundary conditions, and the second kind is obtained by the temporal second-order fast finite compact difference approach for TFGBE with periodic boundary conditions. In the time direction, both schemes employ nonuniform meshes to overcome the initial singularity, where the nonuniform Alikhanov formula with the sum-of-exponentials is used to approximate the time-fractional derivative. As a result, this allows the time direction to achieve second-order accuracy and saves a lot of computational costs. In the space direction, the classical second-order difference formulae are used to discretize the spatial derivatives in the finite difference scheme, which can arrive at second-order accuracy. The developed compact difference formulae are employed to approach the spatial derivatives in the compact difference scheme, which can allow the space direction to achieve fourth-order accuracy. For the two difference schemes, we carry out detailed theoretical analysis, including solvability, boundedness, and convergence analysis. In addition, we provide several numerical examples to test the effectiveness of the proposed fast difference/compact difference schemes and to verify the correctness of the theoretical analysis.