Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion

Using bivariate generating functions, we prove convergence of the Grünwald-Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier-Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak convergence) of the discrete solution towards the probability of sojourn of a diffusing particle. The difference schemes allow also interpretation as discrete random walks. For fractional diffusion with central linear drift we show that in the Fourier-Laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation.