非相对论极限域中非线性狄拉克方程的均匀准确数值方案

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter $0< \varepsilon \ll 1$ inversely proportional to the speed of light. In the massless and nonrelativistic regime, the solution exhibits rapid motion in space and is highly oscillatory in time. Specifically, the wavelength of the propagating waves in time is at $O(\varepsilon)$, while in space it is at $O(1)$ with the wave speed at $O(\varepsilon^{-1}).$ We adopt one leap-frog, t…

Error bounds of fourth-order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

We establish the error bounds of fourth-order compact finite difference (4cFD) methods for the Dirac equation in the massless and nonrelativistic regime, which involves a small dimensionless parameter $0 < \varepsilon \le 1$ inversely proportional to the speed of light. In this regime, the solution propagates waves with wavelength $O(\varepsilon)$ in time and $O(1)$ in space, as well as with the wave speed $O(1/\varepsilon)$ rapid outgoing waves. We adapt the conservative and semi-implicit 4cFD methods to discretize the Dirac equation and rigorously carry out their error bounds depending expli…

非相对论区域下非线性Dirac方程的具有长时间近守恒性的四阶一致精度积分器

Fourth-order uniformly accurate integrators with long time near conservations for the nonlinear Dirac equation in the nonrelativistic regime

针对非相对论区域非线性Dirac方程解的高振荡问题，提出两种结合双尺度方法、指数积分器与算子分解的四阶一致精度积分器，证明其精度与长时间近守恒性，且可扩展至多类Dirac方程，经数值实验验证有效。

非相对论区域下两类非线性耦合色散模型的高阶一致精度全离散格式及其长时间近守恒性

High-order uniformly accurate fully discrete schemes with long time near-conservations for two nonlinear coupled dispersive models in the nonrelativistic regime

针对非相对论区域下两类非线性耦合色散模型，提出结合傅里叶拟谱空间离散与双尺度指数Runge-Kutta时间推进的全离散格式，实现高阶一致精度与长时间近守恒性，数值实验验证了格式的有效性。

Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic regime without magnetic potential

Super-resolution of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter $0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at $O(\varepsilon^2)$. The splitting methods surprisingly show super-resolution, i.e. the methods can capture the solution accurately even if the time step size $τ$ is much larger than the sampled wavelength at $O(\varepsilon^2)$. S…

非相对论Dirac方程时间分裂方法的超分辨

Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

我们建立了狄拉克方程在非相对论极限状态下的Lie-Trotter分裂（$S_1$）和Strang分裂（$S_2$）的误差界，其中小参数$0<\varepsilon\leq 1$与光速成反比.在此极限状态下，溶液在时间上传播具有$O(\varepsilon^2)$波长的波。令人惊讶的是，我们发现分裂方法表现出超分辨率，在这个意义上打破了香农采样定理下的分辨率约束，即即使时间步长$τ$远大于$O(\varepsilon^2)$处的采样波长，该方法也可以准确地捕获解决方案。$S_1$通过建立两个独立的误差界$τ+ \varepsilon$和$τ+ τ/\varepsilon$，证明了$1/2$相对于$\varepsilon$的阶收敛性。此外，如果$τ$是非共振的，即$τ$远离由$\varepsilon$确定的特定区域，则$S_1$将产生改进的均匀一阶$O(τ)$误差界。此外，我们还证明了$S_2$对一般时间步长$τ$以1/2阶速率一致收敛，对非共振时间步长以$3/2$阶速率一致收敛.最后，数值例子来验证我们的研究结果。