一、题目：
Multiscale methods and analysis for the highly oscillatory nonlinear Klein-Gordon equation
二、主讲人：
包维柱
三、摘要：
In this talk, I begin with the nonlinear Klein-Gordon equation (NKGE) under two important parameter regimes, i.e. one is nonrelativistic regime and the other is long-time dynamics with weak nonlinearity or small initial data, while the NKGE is highly oscillatory. I first review our recent works on numerical methods and analysis for solving the NKGE in the nonrelativistic regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the NKGE. Rigorous error estimates show that the EWI spectral method show much better temporal resolution than the FDTD methods for the NKGE in the nonrelativistic regime. In order to design a multiscale method for the NKGE, we establish error estimates of FDTD and EWI spectral methods for the nonlinear Schrodinger equation perturbed with a wave operator. Based on a large-small amplitude wave decomposition to the solution of the NKGE, a multiscale method is presented for discretizing the NKGE in the nonrelativistic regime. Rigorous error estimates show that this multiscale method converges uniformly in spatial/temporal discretization with respect to the small parameter for the NKGE in the nonrelativistic regime. Finally, I discuss issues related error bounds of different numerical methods for the long-time dynamics of NKGE with weak nonlinearity and applications to several highly oscillatory dispersive partial differential equations.
四、主讲人简介：
包维柱教授，1995年博士毕业于清华大学，目前是新加坡国立大学（NUS）数学系教授、研究生研究和学术事务副院长。2013年-2016年为新加坡国立大学Provost讲席教授。2013年获冯康科学计算奖。第26届国际数学家大会（ICM）45分钟特邀报告专家。2022年，包维柱教授当选为美国数学学会（AMS）研究员和SIAM Fellow。担任SIAM Journal of Numerical Analysis等多个国际期刊杂志编委。包维柱教授长期从事科学与工程计算研究，主要工作涉及偏微分方程的数值方法及其在量子物理、流体和材料中的应用。特别在玻色-爱因斯坦凝聚、固态脱湿的建模和模拟以及高振荡偏微分方程的多尺度方法和分析方面做出了重大贡献。
五、邀请人：
蒋晓芸 数学学院教授
六、时间：
5月12日（周四）14:00
七、地点：
腾讯会议
八、联系人：
贾俊青，联系方式：18353112314@163.com
九、主办方：
数学学院
