An implicit finite difference scheme based on the $L2$-$1_σ$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the $L_2$-norm is $\mathcal{O}(τ^2 + h^2)$ with time step $τ$ and mesh size $h$. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.
Study of a fully implicit scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit
Université de Nantes（南特大学）；Laboratoire de Mathématiques Jean Leray, UMR 6629 - CNRS（让·勒雷数学实验室，UMR 6629 - 法国国家科学研究中心）；Université Lille 1（里尔第一大学）；Laboratoire P. Painlevé, CNRS UMR 8524（P. 庞加莱数学实验室，CNRS UMR 8524）；INRIA Lille Nord Europe（欧洲北部法国国家信息与自动化研究所）；Institut de Mathématiques de Toulouse, CNRS UMR 5219（图卢兹数学研究所，CNRS UMR 5219）；Université de Toulouse（图卢兹大学）
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；School of Mathematical Sciences, Queensland University of Technology（昆士兰科技大学理学院）
Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises
研究了可加分数布朗运动驱动的偏微分方程Euler格式的强收敛速度，其中分数布朗运动具有Hurst参数$H\in(\frac13,\frac12)$，漂移系数不要求有界.利用Malliavin演算、粗糙路径理论和$2$ D Young积分，克服了分数阶布朗运动的低正则性和漂移系数的无界性所带来的困难.证明了Euler格式在漂移系数导数有界的情况下具有强$2H$阶，在线性情况下具有强$H+\frac12$阶.数值模拟支持的理论结果。
Fast iterative method with a second order implicit difference scheme for time-space fractional convection-diffusion equations
University of Electronic Science and Technology of China（电子科技大学）；University of Groningen（格罗宁根大学）；Southeast University（东南大学）；Nottingham Trent University（诺丁汉特伦特大学）；Russian Academy of Sciences（俄罗斯科学院）
A fast second-order implicit difference method for time-space fractional advection-diffusion equation
University of Electronic Science and Technology of China（电子科技大学）；Southwestern University of Finance and Economics（西南财经大学）；Neijiang Normal University（内江师范学院）
Relative entropy estimate and geometric ergodicity for implicit Langevin Monte Carlo
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University（上海交通大学数学科学学院、自然科学研究院、教育部线性科学中心）；Department of Mathematics, Duke University（杜克大学数学系）；Department of Physics, Duke University（杜克大学物理系）
本文研究隐式 Langevin Monte Carlo 方法，在漂移仅单边 Lipschitz 条件下，利用 PDE 技术和反射型耦合方法证明了相对熵误差界与 Wasserstein-1 距离下的几何遍历性，并将误差界推广到时间一致情形。
