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

TL;DR

This work addresses time-fractional diffusion-wave equations with high-contrast heterogeneity by applying multicontinuum homogenization to derive coarse-scale macroscopic models. The method formulates constraint cell problems in oversampled RVEs and uses expansions $u\approx \phi_i U_i+\phi_i^m \nabla_m U_i$ to obtain effective coefficients, yielding a strong-form multicontinuum model that accommodates both uniform and mixed time-derivative orders. Numerical experiments in 2D with crossed and layered heterogeneities demonstrate that the multicontinuum models reproduce fine-grid solutions with high accuracy on coarse grids, with errors decreasing as grid resolution and oversampling increase. The approach offers an efficient framework for simulating memory-containing multiscale TFDEs in heterogeneous media, with potential applicability to complex porous, composite, and fractal-like materials. $\frac{\partial^\alpha u}{\partial t^\alpha}-\nabla\cdot(\kappa\nabla u)=f$, $1<\alpha<2$, and the macroscopic models $\widehat{C_{ji}}\frac{\partial^\alpha U_i}{\partial t^\alpha}-\nabla_n(\widehat{B_{ji}^{mn}}\nabla_m U_i)+\frac{1}{\epsilon^2}\widehat{B_{ji}}U_i=f_j$, or the mixed-time-derivative version $\widehat{C_{jip}}\frac{\partial^{\alpha_p} U_i}{\partial t^{\alpha_p}}-\nabla_n(\widehat{B_{ji}^{mn}}\nabla_m U_i)+\frac{1}{\epsilon^2}\widehat{B_{ji}}U_i=f_j$, are validated against fine-scale solutions.

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.

Figures (15)

• Figure 1: Illustration of computation domain $\Omega$, coarse block $\omega$, RVE $R_{\omega}$, and the oversampling domain $R_{\omega}^+$.
• Figure 2: The coefficient $\kappa$ ($\Omega_1$: blue regions; $\Omega_2$: red regions). Crossed field.
• Figure 3: The fine-grid reference solution with $\alpha = 1.5$ at $t=0.1,0.5,1$ for Case 1 in Example 1 (from left to right).
• Figure 4: Average solution with $\alpha=1.5$ at $t=0.1,0.5,1$ for Case 1 in Example 1 (from left to right), $H = 1/40$. First row: reference averaged solution in $\Omega_1$. Second row: multiscale solution in $\Omega_1$.
• Figure 5: Average solution with $\alpha=1.5$ at $t=0.1,0.5,1$ for Case 1 in Example 1 (from left to right), $H = 1/40$. First row: reference averaged solution in $\Omega_2$. Second row: multiscale solution in $\Omega_2$.