报告题目：Recent developments and challenges on nonuniform approximations of Caputo's derivative
个人介绍：1998年本科毕业于空军气象学院大气科学系，2001年硕士毕业于解放军理工大学应用数学与物理系，2010年博士毕业于东南大学数学系。学术研究方向为偏微分方程数值解与可扩展数值并行算法。目前主要研究分数阶微分方程的非均匀离散逼近，在《SIAM Journal on Numerical Analysis》,《Journal of Computational Physics》,《Journal of Scientific Computing》,《中国科学》等国内外专业期刊上发表学术论文20余篇，被引用300余次。
Time-fractional linear and nonlinear parabolic equations involving the fractional Caputo derivative are important models in modeling complex systems such as glassy and distorted materials. In developing numerical methods especially for fractional partial differential equations, the Caputo’s derivative introduces some new difficulties, such as initial singularity, kernel singularity and historical memory.
In this talk, we report the L1 and Alikhanov formula on general nonuniform time grids for solving fractional linear and nonlinear reaction- subdiffusion equations. By introducing a discrete convolution kernel of Riemann-Liouville fractional integral, we establish a discrete fractional Gronwall-type inequality, which is a generalization of classical Gronwall inequality for local time derivative and is useful in stability and convergence analysis of numerical schemes. To simplify the consistence analysis of discrete Caputo formulas, we bound the local truncation errors by a class of discrete convolution forms, and consider a global convolution error with the discrete Riemann- Liouville integral kernel. This novel technique avoids the detailed evaluations of discrete convolution kernels and provides a simple way to analyze the approximate errors of discrete Caputo formula. For the fractional linear reaction-subdiffusion equation, new L^2-norm and H^1-norm error estimate reflecting the regularity of solution is obtained for simple discrete schemes on nonuniform time meshes.
To accelerate the time-integration, a fast L1 formula on nonuniform meshes is constructed by splitting the Caputo derivative into a local part and a history part, and applying the sum-of-exponentials technique to approximate the singular kernel over the history interval. We construct a two-level fast linearized algorithm for two-dimensional time-fractional semilinear reaction-subdiffusion equation. The resulting scheme is computationally efficient in long-time simulations since it requires only $O(M\log N)$ storage and $O(MN\log N)$ computational cost with $M$ grid points in space. Unconditionally maximum norm error estimate reflecting the regularity of solution is established by applying discrete $H^2$ energy method, discrete fractional Gronwall-type inequality and global consistence error analysis. Numerical tests are provided to confirm the sharpness of error analysis.
Some open problems and challenges are also reported in this talk.