An exponential integrator multicontinuum homogenization method for fractional diffusion problem with multiscale coefficients

In this paper, we present a robust and fully discretized method for solving the time fractional diffusion equation with high-contrast multiscale coefficients. We establish the homogenized equation using a multicontinuum approach and employ the exponential integrator method for time discretization. The multicontinuum upscaled model captures the physical characteristics of the solution for the high-contrast multiscale problem, including averages and gradient effects in each continuum at the coarse scale. We then use the exponential integration method for the nonlocal time fractional derivative and it can handle semilinear problem in an efficient way. Convergence analysis of the numerical scheme is provided, along with illustrative numerical examples. Our results demonstrate the accuracy, efficiency, and improved stability for varying order of fractional derivatives.