Multicontinuum Modeling of Time-Fractional Diffusion-Wave Equation in Heterogeneous Media

TL;DR

This paper develops a multicontinuum homogenization approach to efficiently model and solve time-fractional diffusion-wave equations in heterogeneous media with high contrast, addressing computational challenges of multiscale problems.

Contribution

It introduces a novel multicontinuum homogenization method for time-fractional diffusion-wave equations in heterogeneous media, enabling accurate multiscale modeling.

Findings

Numerical results verify the accuracy of the multicontinuum model.

The approach effectively captures multiscale effects in heterogeneous media.

The method reduces computational complexity for high-contrast problems.

Abstract

This paper considers a time-fractional diffusion-wave equation with a high-contrast heterogeneous diffusion coefficient. A numerical solution to this problem can present great computational challenges due to its multiscale nature. Therefore, in this paper, we derive a multicontinuum time-fractional diffusion-wave model using the multicontinuum homogenization method. For this purpose, we formulate constraint cell problems considering various homogenized effects. These cell problems are implemented in oversampled regions to avoid boundary effects. By solving the cell problems, we obtain multicontinuum expansions of fine-scale solutions. Then, using these multicontinuum expansions and supposing the smoothness of the macroscopic variables, we rigorously derive the corresponding multicontinuum model. Finally, we present numerical results for two-dimensional model problems with different time-fractional derivatives to verify the accuracy of our proposed approach.