Tianyuan Mathematical Center
in Southwest China

Stability of high order IMEX and EIN numerical methods for high order PDEs

报告题目：Stability of high order IMEX and EIN numerical methods for high order PDEs

报告专家：成娟 研究员（北京应用物理与计算数学研究所）

报告时间：2022年12月5日 16:00—17:00

报告地点：腾讯会议：710-637-116（无密码）

报告摘要：Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k ≥ 2) order PDEs, the explicit method may suffer from a severe time step restriction Δt = O(Δxk) for stability. Implicit methods are generally unconditionally stable, however, they are cumbersome for nonlinear equations, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The implicit-explicit (IMEX) methods, which treat the stiffer terms implicitly and the rest of the terms explicitly, can not only alleviate time step constraint, but also reduce the difficulty of solving the algebraic system especially when the stiffer terms are linear. We have analyzed the stability of various large time-stepping IMEX schemes for the high order PDEs such as the convection-dispersion equation in conjunction with high order finite difference method for spatial discretization.

Furthermore, for the equations with nonlinear high derivative terms, the IMEX methods are still too expensive to use. A better alternative is to use the explicit-implicit-null (EIN) method. The EIN method does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we have discussed the stability of high order finite difference and the LDG schemes for solving the convection-dispersion and the bi-harmonic type equations, respectively. Our main contribution is to show rigorously that the resulting numerical schemes are stable for large time step if stabilization terms of appropriate size are chosen. Numerical experiments are given to assess accuracy and stability.

专家简介：成娟，北京应用物理与计算数学研究所研究员。主要从事可压缩流体力学、辐射输运、辐射流体力学等对流占优偏微分方程的高精度、健壮数值方法研究。现担任“Journal of Computational Physics”期刊编委、北京计算数学学会副理事长、中国工业与应用数学学会（CSIAM）竞赛工作委员会副主任、中国数学会计算数学专业委员会理事等。主持国家自然科学基金重点项目。曾获航空航天工业部科技进步奖二等奖与中国工程物理院科技创新奖二等奖。应邀在“第十七届CSIAM年会”等国内外学术会议上作大会报告。

邀请人：贺巧琳