Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods are presented and analyzed for the long-time dynamics of the Dirac equation with small potentials characterized by $\varepsilon \in (0, 1]$ a dimensionless parameter. Based on the (symmetric) exponential wave integrator for temporal derivatives in phase space followed by applying the Fourier pseudospectral discretization for spatial derivatives, the EWI-FP methods are explicit and of spectral accuracy in space and second-order accuracy in time for any fixed $\varepsilon = \varepsilon_0$. Uniform error bounds are rigorously carried out at $O(h^{m_0}+\tau^2)$ up to the time at $O(1/\varepsilon)$ with the mesh size $h$, time step $\tau$ and $m_0$ an integer depending on the regularity of the solution. Extensive numerical results are reported to confirm our error bounds and comparisons of two methods are shown. Finally, dynamics of the Dirac equation in 2D are presented to validate the numerical schemes.
翻译:展示了两种指数波混集器 Fourier伪光谱(EWI-FP)方法,并分析Dirac方程式的长期动态,其潜力很小,以美元为0,1美元为无维参数。根据对空间空间中的时间衍生物的指数波混集器,然后对空间衍生物应用四光谱分解法,EWI-FP方法十分清晰,空间光谱精度和空间次等精度准确度,对于任何固定的 $ = varepsilon = varepsilon_0美元为第二阶次的精确度。最后,对2Dirac方程式的动态进行了2D的验证。