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．

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.