基于非线性分数阶微分方程的quenching问题数值模拟研究

由于quenching问题的解关于时间的导数在有限时间内会发生奇性，故其一直是复杂热物理领域研究的热点和难点。研究表明分数阶微分方程在反常扩散等领域比整数阶微分方程能更准确描述实际问题。鉴于目前分数阶微分方程应用于quenching问题大都针对一维情形采用低精度数值方法，对quenching现象的数值模拟不够高效精确。本项目拟对三类带有奇异点的非线性分数阶奇异反应扩散方程，基于时间和空间自适应网格算法，结合算子分裂技术建立半离散、全离散高精度有限差分格式，从一维推广到高维，证明算法的能量守恒性、收敛性及误差分析，对此三类方程的quenching现象前后解的渐进变化进行数值模拟，精确捕捉其临界区间、初始quenching时间和空间位置，讨论分数阶导数对其临界值的影响，与经典quenching问题比对验证此数值模拟的可靠性。此项目有助于揭示quenching现象的内在机理，优化燃烧过程。

Because the derivatives of the solutions of the quenching problem with time have singularity in a limited time, it has been one of the difficulties and hot spots of complex thermophysical researches. Studies have shown that fractional differential equations can describe some practical problems more accurately than integer-order differential equations in some areas such as anomalous diffusion etc. Most existing research works focus primarily on one-dimensional problems adopting lower order numerical methods. The numerical simulations of the quenching phenomena are less efficient and accurate. In this project, we plan to aim at three mathematical models of the quenching problem based on the latest three kinds of fractional nonlinear singular reaction-diffusion equations. Our investigations will begin with one-dimensional problems and then to extend to high-dimensional processes. Challenging issues including high-order semi and full-discretization, energy conservation, error conservation, error analysis and operator splitting will be addressed together with my graduate students and collaborators. Combustion time, location and occurring criterion will be investigated and analyzed. Rigorous asymptotic and numerical analysis will be carried out. We focus on revealing the relationship between fractional derivative and the quenching critical values and verifying the reliability of numerical simulation by comparison with the classical quenching problem. This project will help to reveal the underlying mechanism of the quenching problem and optimize the combustion process.