Numerical stabilization for a mixture system with kind damping

1. Problem Statement

The paper investigates the asymptotic behavior and numerical stabilization of a system modeling the dynamics of a mixture of two rigid solids with porosity. The specific focus is on the interaction between the porosity field ($u$) and the temperature field ($v$) within a homogeneous, isotropic elastic body.

The governing equations are a coupled system of wave equations where the standard viscous damping terms (typically proportional to $\Delta u_t$) are replaced by fractional derivative damping terms of the Caputo type, denoted as $\partial^{\alpha, \eta}_t u$ and $\partial^{\alpha, \eta}_t v$. The system is defined as:

$$\begin{cases} \rho_1 u_{tt} - a_{11}\Delta u - a_{12}\Delta v + \alpha(u-v) + \partial^{\alpha, \eta}_t u = 0 \\ \rho_2 v_{tt} - a_{12}\Delta u - a_{22}\Delta v - \alpha(u-v) + \partial^{\alpha, \eta}_t v = 0 \end{cases}$$

subject to homogeneous Dirichlet boundary conditions and specific initial conditions. The parameters $\alpha > 0$ represents the coupling, while the fractional derivative order satisfies $0 < \alpha < 1$ and $\eta \geq 0$.

The primary mathematical challenge is that fractional derivatives in time are non-local operators, making direct analysis and numerical simulation difficult. The authors aim to:

1. Reformulate the system to eliminate the fractional derivative.
2. Prove the well-posedness and stability (strong and polynomial) of the solutions.
3. Develop a numerical scheme to verify the theoretical decay rates.

2. Methodology

A. Augmented Model (Analytical Approach)

To handle the non-local nature of the fractional derivative, the authors employ an augmentation technique based on the work of Choi and MacCamy and others.

• Transformation: They introduce auxiliary variables $\phi_1(x, t, \xi)$ and $\phi_2(x, t, \xi)$ defined over an extended domain $\Omega \times (0, \infty) \times \mathbb{R}$.
• Diffusion Equation: The fractional derivative $\partial^{\alpha, \eta}_t u$ is represented as an integral of the solution to a diffusion-like equation:
  $$\phi_t + (\xi^2 + \eta)\phi = \mu(\xi)u_t$$
  where $\mu(\xi) = |\xi|^{(2\alpha-1)/2}$.
• Resulting System: The original system is transformed into a coupled system of wave equations and parabolic equations (in the auxiliary variable $\xi$). This allows the use of standard semigroup theory.

B. Semigroup Theory and Stability Analysis

• Well-posedness: The authors define a suitable Hilbert energy space $\mathcal{H}$ and an unbounded linear operator $A$. They prove that $A$ generates a $C_0$-semigroup of contractions, ensuring the existence and uniqueness of solutions.
• Strong Stability: Using the Arendt-Batty theorem, they prove that the semigroup is strongly stable (solutions converge to zero as $t \to \infty$). This relies on showing that the operator $A$ has no purely imaginary eigenvalues and that the intersection of its spectrum with the imaginary axis is countable.
• Polynomial Stability:
  • For $\eta > 0$, they establish a decay rate of $E(t) \leq C t^{-\frac{1}{1-\alpha}}$.
  • For $\eta = 0$, they utilize spectral analysis results (Borichev-Tomilov type theorems) to derive decay rates based on the behavior of the resolvent $(i\lambda I - A)^{-1}$ near zero and infinity.

C. Numerical Approximation

The authors develop a numerical scheme to validate the theoretical decay rates:

• Spatial Discretization: A Finite Volume Method (FVM) is used for the spatial variables $u$ and $v$ on a 2D domain. The Laplacian is approximated using central finite differences.
• Fractional Term Approximation: The auxiliary diffusion equations for $\phi$ are discretized using the Crank-Nicolson method to preserve energy conservation properties. The integral defining the fractional derivative is approximated via a quadrature rule over the auxiliary variable $\xi$.
• Time Discretization: A $\beta$-Newmark scheme is employed for the wave equations to ensure second-order accuracy in time.
• Energy Conservation: The discrete energy is shown to decay monotonically, consistent with the continuous model.

3. Key Contributions

1. Augmented Formulation: Successfully reformulated a mixture system with fractional damping into an augmented system involving a continuous spectrum of diffusion equations, facilitating rigorous spectral analysis.
2. Stability Proofs: Provided rigorous proofs for strong stability and polynomial stability for both cases $\eta > 0$ and $\eta = 0$. They explicitly quantified the decay rates, showing dependence on the fractional order $\alpha$.
3. Numerical Validation: Developed a robust numerical scheme combining FVM, Crank-Nicolson, and Newmark methods. This scheme successfully captures the non-exponential (polynomial) decay of energy, which is often difficult to resolve numerically due to the presence of high-frequency modes near the imaginary axis.
4. Spectral Analysis: Performed numerical spectral analysis of the discretized operator, confirming that eigenvalues lie in the left half-plane and cluster near the imaginary axis, explaining the slow (polynomial) decay rather than exponential decay.

4. Results

• Theoretical Decay Rates:
  • The energy $E(t)$ decays polynomially.
  • If $\eta > 0$, the decay is $O(t^{-\frac{1}{1-\alpha}})$.
  • If $\eta = 0$, the decay is also polynomial but follows a different rate derived from the resolvent growth conditions.
• Numerical Simulations:
  • Example 1: Demonstrated that without the dissipative term, energy is conserved (constant), while with the fractional term, energy decays.
  • Example 2 (Spectrum): Visualized the spectrum of the operator $A$. Eigenvalues were found to be close to the imaginary axis, confirming the lack of exponential decay (which would require eigenvalues bounded away from the imaginary axis).
  • Example 3 (Long-time Decay): Simulations with non-smooth initial data (to excite high frequencies) confirmed the polynomial decay behavior. Log-log plots of energy vs. time showed linear trends consistent with the theoretical rates $1/t$ and $1/t^2$ (depending on parameters), validating the analytical predictions.

5. Significance

• Modeling Real-World Materials: The model is highly relevant for describing viscoelastic bituminous mixtures used in pavement systems, where fractional derivatives accurately capture creep and recovery phenomena that integer-order models cannot.
• Bridging Theory and Computation: The paper provides a critical link between abstract functional analysis (semigroups, fractional calculus) and practical numerical simulation. It demonstrates that complex fractional damping systems can be effectively simulated using augmented models.
• Insight into Decay Mechanisms: The work clarifies that fractional damping leads to polynomial decay rather than the exponential decay typical of standard viscous damping. This is crucial for engineering applications where long-term stability predictions are required; assuming exponential decay could lead to overestimating the system's stability.
• Numerical Robustness: The proposed scheme handles the non-local nature of fractional derivatives efficiently by converting them into local PDEs in an extended domain, offering a template for simulating other fractional-order systems.