求解时间分数阶动-不动区输运方程的数值方法
A numerical approach for solving the time-Fractional Mobile-Immobile Transport Equation
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中文总结 AI 辅助
针对时间分数阶动-不动区输运方程，提出两种全离散数值格式，经理论分析与数值验证，可高效准确模拟多孔介质中的反常输运现象。
中文摘要 AI 辅助
分数阶扩散模型为描述非均质多孔介质中的反常输运现象提供了强大框架。动-不动区模型是表征此类反常扩散的基础方法，专门用于解决由可动区域与不动区域间的质量传递导致的溶质输运延迟问题。本研究针对由阶数α∈(0,1)的Caputo导数控制的一维时间分数阶动-不动区模型，开发并分析了两种全离散数值格式。空间离散采用非对称内罚间断Galerkin方法，时间导数分别用Crank-Nicolson L1和L2-1σ公式近似，得到两种具有高阶精度的不同格式。研究建立了严格的稳定性与误差分析，结果表明该方法的最优收敛率取决于精确解的正则性。数值实验验证了理论预测，证明了所提格式的高效性与鲁棒性。该框架为模拟多孔介质及相关领域的复杂分数阶输运过程提供了可靠且准确的方法。
英文摘要
Fractional diffusion models provide a powerful framework for describing anomalous transport phenomena in heterogeneous porous media. The Mobile-Immobile model is a fundamental approach for characterizing such anomalous diffusion, specifically addressing the delayed solute transport caused by mass transfer between mobile and immobile regions. In this study, we develop and analyze two fully discrete numerical schemes for the one-dimensional time-fractional Mobile-Immobile model governed by the Caputo derivative of order $α\in (0,1)$. The spatial discretization is carried out using a non-symmetric interior penalty discontinuous Galerkin method, while the temporal derivative is approximated by the Crank-Nicolson L1 and L2-$1_σ$ formulas, resulting in two distinct schemes with high-order accuracy. Rigorous stability and error analyses are established, showing that the methods achieve optimal convergence rates depending on the regularity of the exact solution. Numerical experiments verify the theoretical predictions and demonstrate the efficiency and robustness of the proposed schemes. The presented framework provides a reliable and accurate approach for simulating complex fractional transport processes in porous media and related fields.
