一种用于模拟双侧分数阶对流-扩散淬火问题的半自适应有限差分方法
A semi-adaptive finite difference method for simulating two-sided fractional convection-diffusion quenching problems
AI总结：
本文提出了一种半自适应有限差分方法，用于模拟一维双侧 Riemann-Liouville 分数阶对流-扩散问题的淬火现象，通过加权平均近似和 Grünwald 公式离散，分析了格式的守恒性、单调性、正性和稳定性，并探讨了分数阶对淬火特性的影响。
摘要：
本文研究了一维双侧 Riemann-Liouville 分数阶对流-扩散问题的淬火解。分数阶空间导数采用加权平均近似结合标准和移位 Grünwald 公式进行离散。对流项使用简单的 Euler 公式处理，得到半离散非线性常微分方程组。严格证明了该格式的守恒性，并通过模拟实验进行了验证。研究进一步推进到全离散半自适应有限差分方法。对格式的单调性、正性和稳定性进行了详细分析。研究评估了分数阶对淬火位置、淬火时间和临界长度的潜在影响。对计算结果进行了充分讨论和分析，为双侧分数阶对流-扩散问题建模的淬火现象提供了更全面的理解。
英文摘要：
This paper investigates quenching solutions of an one-dimensional, two-sided Riemann-Liouville fractional order convection-diffusion problem. Fractional order spatial derivatives are discretized using weighted averaging approximations in conjunction with standard and shifted Grünwald formulas. The advective term is handled utilizing a straightforward Euler formula, resulting in a semi-discretized system of nonlinear ordinary differential equations. The conservativeness of the proposed scheme is rigorously proved and validated through simulation experiments. The study is further advanced to a fully discretized, semi-adaptive finite difference method. Detailed analysis is implemented for the monotonicity, positivity and stability of the scheme. Investigations are carried out to assess the potential impacts of the fractional order on quenching location, quenching time, and critical length. The computational results are thoroughly discussed and analyzed, providing a more comprehensive understanding of the quenching phenomena modeled through two-sided fractional order convection-diffusion problems.
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