A numerical approach for solving the time-Fractional Mobile-Immobile Transport Equation

Technical Summary: A Numerical Approach for Solving the Time-Fractional Mobile-Immobile Transport Equation

Problem Statement
The paper addresses the numerical solution of the one-dimensional time-fractional Mobile-Immobile (MIM) transport equation, a model used to describe anomalous diffusion in heterogeneous porous media. Unlike classical Fickian models, the MIM model accounts for delayed solute transport caused by mass transfer between mobile and immobile regions, governed by a Caputo fractional derivative of order $\alpha \in (0, 1)$. The specific initial-boundary value problem (IBVP) considered is:
$$\lambda_1 \frac{\partial u}{\partial t} + \lambda_2 {}^C D^\alpha_{0,t} u + Lu = f(x,t),$$
where $L$ is a spatial elliptic operator, and the system is subject to zero Dirichlet boundary conditions. The primary challenge lies in the nonlocal nature of the fractional derivative and the need for high-order accuracy in both space and time to capture memory effects and anomalous transport behaviors efficiently.

Methodology
The author develops two fully discrete numerical schemes that combine spatial and temporal discretizations within a unified framework:

1. Spatial Discretization: The Non-Symmetric Interior Penalty Discontinuous Galerkin (NIPG) method is employed. This approach utilizes a broken Sobolev space and allows for discontinuities across element boundaries. The spatial operator is approximated using a bilinear form $B_h(\cdot, \cdot)$ that includes a penalty term to ensure stability and coercivity.
2. Temporal Discretization: Two distinct strategies are proposed for approximating the time derivatives (both the first-order and the Caputo fractional derivative):
   • Scheme 1 (CN-L1-NIPG): Combines the NIPG method with a Crank-Nicolson (CN) framework applied to the standard L1 formula for the Caputo derivative. This scheme targets $(2-\alpha)$-order temporal accuracy.
   • Scheme 2 (L2-1$\sigma$-NIPG): Combines the NIPG method with the L2-1$\sigma$ formula (introduced by Alikhanov) for the Caputo derivative, also within a CN framework. This scheme aims for second-order temporal accuracy.

The analysis relies on rigorous stability proofs using energy norms and error estimates derived via elliptic projections and discrete Grönwall-type inequalities.

Key Contributions

• Novel Schemes: The paper introduces two distinct fully discrete formulations (CN-L1-NIPG and L2-1$\sigma$-NIPG) specifically for the time-fractional MIM model, a combination that has received limited attention in existing literature compared to simpler fractional diffusion equations.
• Theoretical Analysis:
  • Stability: Unconditional stability is established for both schemes. The analysis demonstrates that the numerical solutions remain bounded by the initial data and source terms, independent of the mesh size.
  • Convergence: Optimal error estimates are derived. The CN-L1-NIPG scheme achieves an error bound of $O(\tau^{2-\alpha} + h^{k+1})$, while the L2-1$\sigma$-NIPG scheme achieves $O(\tau^2 + h^{k+1})$, where $\tau$ is the time step, $h$ is the spatial mesh size, and $k$ is the polynomial degree. These rates depend on the regularity of the exact solution.
• Comprehensive Error Bounds: The error analysis explicitly accounts for the regularity of the solution, providing a theoretical foundation for the observed convergence rates.

Results
Numerical experiments were conducted on a test problem with a known exact solution to validate the theoretical predictions.

• CN-L1-NIPG: The numerical results confirmed the theoretical convergence order of $(2-\alpha)$ in the temporal direction for various values of $\alpha \in (0, 1)$.
• L2-1$\sigma$-NIPG: The experiments demonstrated that this scheme achieves second-order temporal accuracy, outperforming the L1-based scheme in terms of convergence rate for the time derivative.
• Robustness: Both schemes demonstrated efficiency and robustness across different fractional orders and mesh refinements, with the L2-1$\sigma$ method showing superior temporal accuracy.

Significance and Claims
The paper claims to provide a reliable and accurate computational framework for simulating complex fractional transport processes in porous media. By combining the flexibility of the NIPG method with high-order temporal discretizations, the proposed schemes offer a unified approach to modeling non-Fickian transport phenomena. The author asserts that their work fills a gap in the literature regarding high-order numerical methods for the MIM model, offering a tool that is both theoretically sound and practically efficient for applications in hydrology, contaminant transport, and sediment dispersion. The study concludes that these methods pave the way for future extensions to higher-dimensional and variable-order fractional models.