A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative

In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For $0<\alpha (t)<1$, we prove that the method is unconditionally stable and the errors attain $(k+1)$-th order of accuracy for piecewise $P^k$ polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.