Session I.5 - Geometric Integration and Computational Mechanics
Poster
Error Estimates of Numerical Methods for the Long-time Dynamics of the Nonlinear Klein-Gordon Equation
Yue Feng
Sorbonne University, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionless parameter. Different numerical methods are adopted to discretize the NKGE and rigorous error bounds are established for the long-time dynamics. Numerical methods include finite difference methods, exponential wave integrators and time-splitting methods with particular attentions paid on error bounds of different numerical methods explicitly depending on the mesh size $h$, time step $\tau$ as well as the parameter $\varepsilon$ up to the time $t= T/\varepsilon^2$ with $T \gt 0$ fixed. As a by-product, our results are extended to an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^2)$ in time. Extensive numerical examples are provided to confirm our error bounds and demonstrate that they are sharp.
Joint work with Weizhu Bao (National University of Singapore), Yongyong Cai (Beijing Normal University), Chunmei Su (Tsinghua University) and Wenfan Yi (Hunan University).