报告题目：Alikhanov's $L2$-$1_\sigma$ scheme on fitted meshes for time-fractional initial-value problems and initial-boundary-value problems.
报告摘要: Alikhanov's high-order $L2$-$1_\sigma$ scheme for Caputo fractional derivatives of order $\alpha\in (0,1)$ is generalised to nonuniform meshes and analysed for initial-value problems (IVPs) and initial-boundary value problems (IBVPs) whose solutions display a typical weak singularity at the initial time. It is shown that, when the mesh is chosen suitably, the scheme attains order $3-\alpha$ convergence in time for the 1-dimensional IVP and second-order convergence in time for the IBVP. For the IBVP we consider the case where the spatial domain is the unit square and use a spectral method to discretise in space, but other spatial domains and other spatial dimensions and discretisations are possible. Numerical results demonstrate the sharpness of the theoretical convergence estimates. An improved discretisation for the IBVP attains order $3-\alpha$ convergence (the same order as the IVP) in numerical experiments, but its analysis remains open.
报告人简介: Martin Stynes is a Full Professor at Beijing Computational Science Research Center since 2014, supported by China’s 1000 Talents Foreign Experts Program. Before then, he worked at University College Cork, Ireland for almost 30 years. For many years his research focussed on numerical methods for convection-dominated problems — the book “Robust Numerical Methods for Singularly Perturbed Differential Equations”by Roos, Stynes & Tobiska is the standard reference work in this area. In recent years he has concentrated on the analysis and numerical solution of fractional-derivative differential equations.
