A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations

In this paper, we investigate the well-posedness and solution regularity of a variable-order time-fractional diffusion equation, which is often used to model the solute transport in complex porous media where the micro-structure of the porous media may changes over time. We show that the variable-order time-fractional diffusion equations have flexible abilities to eliminate the nonphysical singularity of the solutions to their constant-order analogues. We also present a finite volume approximation and provide its stability and convergence analysis in a weighted discrete norm. Furthermore, we develop an efficient parallel-in-time procedure to improve the computational efficiency of the variable-order time-fractional diffusion equations. Numerical experiments are performed to confirm the theoretical results and to demonstrate the effectiveness and efficiency of the parallel-in-time method.