报告题目1：Sharp error estimate of BDF2 scheme with variable time steps for linear reaction-diffusion equations

报告人：张继伟教授

报告时间：2021年3月23日（星期二）14：30-18：00

报告地点：E-2028

报告摘要：While the variable time-steps backward differentiation formula (BDF2) is valuable and widely used to capture the multi-scale dynamics of model solutions, however, the stability and convergence of BDF2 with variable time steps still remain incomplete. In this work, we revisit BDF2 scheme for linear diffusion reaction problem. By introducing the conception of the discrete complementary convolution (DCC) kernels and the positive semi-definiteness of BDF2 convolution kernels, we present that BDF2 scheme is unconditionally stable under the restriction condition of adjacent time-step ratios: $r_k:= \leq r_{\max} = 4.8645. $ With the use of DCC kernels, the second-order temporal convergence can be achieved under this new ratios. Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time steps for the first level solution $u^1$. This is, the first-order consistence of the first level solution $u^1$ like BDF1 (i.e. Euler scheme)} as a starting point does not cause the loss of global temporal accuracy, and the ratios are updated to $r_k \leq 4.8645. $

报告题目2：The optimal second-order convergence of BDF2 scheme for molecular beam expitaxial models without slop selection

报告人：张继伟教授

报告时间：2021年3月23日（星期二）19：30-21：30

报告地点：E-2028

报告摘要：The stability and convergence of two-step backward differentiation formula (BDF2) with variable time steps still remain incomplete for solving the molecular beam epitaxial model without slope selection. In this talk, we first prove the proposed BDF2 to preserve a modified energy dissipation law under a new adjacent time-step radios: $r_k:= 4.8645$. After that, we introduce the recently developed techniques of the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and present the sharp second-order convergence of the BDF2 scheme with the new adjacent time-step radios: $r_k:=\tau_k/\tau_{k-1}\leq 4.8645 -\delta$, where $\delta>0$ is a given any small constant. Our analysis shows that (1) the convergence will remain the second-order under the new adjacent time-step ratios; (2) the first-order consistency scheme for the start step such as BDF1 is enough to ensure the globally optimal convergence order.

个人简介：张继伟，武汉大学数学与统计学院教授，博士生导师。2003和2006年在郑州大学获得学士和硕士学位，2009年在香港浸会大学获得博士学位。随后在南洋理工大学和纽约大学克朗所从事博士后研究，2014年5月在北京计算科学研究中心工作，2018年11月到武汉大学工作。主要研究领域包括偏微分方程和非局部模型的数值解法，以及神经科学的建模与计算。主要成果发表在SIAM Journal on Scientific Computing, SIAM Journal on Numerical Analysis, Mathematics of Computation， Journal of Computational Neuroscience, Plos Comput. Bio等国际知名期刊上。

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2021年3月22日