论文标题
具有改进的非线性klein-gordon方程的对称集成符，在非依次限制方程
Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
论文作者
论文摘要
在本文中，我们与具有无量纲参数$ 0 <\ varepsilon \ ll 1 $的非线性相对论klein-gordon（NRKG）方程的对称积分器有关，这与光速成反比。该模型时间的高度振荡属性对应于参数$ \ varepsilon $，当$ \ eps $很小时，方程式具有很强的非线性。在设计数值方法时，有两个方面带来了显着的数值负担。我们提出和分析了一类新型的对称积分器，该集成剂基于对该问题的某些公式方法，傅立叶伪 - 光谱方法和指数积分器。通过使用所提出的对称属性和隐式指数积分器的硬命令条件来构建两个实用的集成器，直到四个订单。严格研究了获得的集成器的收敛性，并且表明时间的准确性已提高为$ \ natercal {o}（\ varepsilon^{3} {3} \ hh^2）$和$ \ Mathcal {o} o}（\ varepsilon^{\ varepsilon^{4} {4} \ hh hh^4）$ for Time Stestep。长时间内，通过使用调制傅立叶扩展为多阶段集成商建立了近节能量。即使在方案中使用了大的步骤，这些理论结果也是可以实现的。 NRKG方程式上的数值结果表明，所提出的积分器提高了统一的误差界限，出色的长时间能量节省和竞争效率。
In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when $\eps$ is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be $\mathcal{O}(\varepsilon^{3} \hh^2)$ and $\mathcal{O}(\varepsilon^{4} \hh^4)$ for the time stepsize $\hh$. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency.
