论文标题
正弦方程的长期动力学的Lawson型指数积分器上的改进统一误差界限
Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine--Gordon equation
论文作者
论文摘要
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \varepsilon \ll 1$ a dimensionless parameter up to the time at $ O（1/\ Varepsilon^2）$。数值方案将Lawson型指数积分器与傅立叶型积分器与空间离散化的傅立叶伪谱法结合在一起，这要归功于快速的傅立叶变换，在实用计算中完全显式且有效。通过将线性部分与正弦函数分开并采用规律性补偿振荡（RCO）技术，该技术被引入以通过相位取消来处理多项式非线性，我们在$ o（\ varepsilon^2τ）$ o（cy（cy）$ o（cy（AT）$ o（cy）中，我们执行了改进的误差范围。 $ o（H^M+\ Varepsilon^2τ）$用于全滴之前，直到$ t _ {\ varepsilon} = t/\ varepsilon^2 $，$ t> 0 $固定。这是第一项建立针对具有非多功能非线性的NKGE长期动力学结合的改进的均匀误差的工作。绑定的改进错误将扩展到$ o（\ varepsilon^2）的振动正弦方程，时间为$ o（\ varepsilon^{ - 2}）$ wave速度，这表明时间误差与$ \ varepsilon $相关时，$ o o o o（aS $ o o o（$ o o o o o o o o o o o o o o o o o o o o o o o o o o o \ varepsieloys $）最后，显示数值示例可以确认改进的误差界限并证明它们是锋利的。
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \varepsilon \ll 1$ a dimensionless parameter up to the time at $O(1/\varepsilon^2)$. The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discreization at $O(\varepsilon^2τ)$ instead of $O(τ)$ according to classical error estimates and at $O(h^m+\varepsilon^2τ)$ for the full-discretization up to the time $T_{\varepsilon} = T/\varepsilon^2$ with $T>0$ fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the NKGE with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with $O(\varepsilon^2)$ wavelength in time and $O(\varepsilon^{-2})$ wave speed, which indicates that the temporal error is independent of $\varepsilon$ when the time step size is chosen as $O(\varepsilon^2)$. Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.
