分数阶Cattaneo方程高精度有限差分法的研究

In recent years, fractional partial differential equations have been widely applied in many fields, such as signal processing, quantum economic, hydrological geography, et al. The Cattaneo model is used to describe the anomalous diffusion and the non-Fourier heat conduction phenomena in the nature. The goal of this project is to construct high order finite difference schemes for fractional Cattaneo equations. High order approximations are used to discretize the Caputo fractional derivative, the compact operator for the spatial second order derivative is adopted to raise the convergence accuracy, for the Riemann-Liouville fractional derivative, the fourth order scheme is employed. As to the numerical scheme with nonsymmetric structure, this project intends to use the energy method combining with the matrix decomposition technique to ensure the stability and convergence.

近年来，分数阶偏微分方程在信号处理、量子经济和水文地理等领域都得到广泛的应用．Cattaneo本构关系模型是用来描述自然界中反常扩散现象和非Fourier热传导现象的模型．本项目拟对几类分数阶Cattaneo方程构建高精度差分格式，时间Caputo分数阶导数主要采用申请者及其他学者提出的高阶格式，空间二阶导数借助紧算子来提高收敛精度，空间Riemann-Liouville分数阶导数则采用四阶离散公式来构建高精度差分格式．对具有非对称结构的数值格式，本项目拟采用能量法与矩阵分解技巧相结合来保证格式的稳定性和收敛性．

该基金项目主要研究分数阶微分方程的高效数值解法，在该项目的资助下，项目组主要进行了如下研究工作：（1）对带Caputo分数阶导数的两点边值问题构建了高精度的有限差分格式，严格保证了格式的无条件稳定性和收敛性，并进行数值实验来验证理论结果，这里模型解析解的二阶导数是无界的；（2）我们建立了黎曼边界条件下带空间四阶导数的分数阶慢扩散方程的高精度紧格式，成功克服了黎曼边界条件带来的困难，严格给出了无条件稳定和收敛性证明，并进行数值实验来验证理论结果；（3）我们还研究了时间分数阶Cattaneo方程的高精度快速算法，论文正在做最后润色，等待投稿。