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A parareal finite volume method for variable-order time-fractional diffusion equations. (English) Zbl 07262120
Summary: 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.
##### MSC:
 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65Y05 Parallel numerical computation 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals 76S05 Flows in porous media; filtration; seepage 35Q35 PDEs in connection with fluid mechanics
FODE
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