[Paper Review] Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic limit regime.

Weizhu Bao, Yongyong Cai+1 more|arXiv (Cornell University)|Jun 25, 2019

Electromagnetic Simulation and Numerical Methods42 references4 citations

TL;DR

This paper analyzes the Lie-Trotter ($S_1$) and Strang ($S_2$) splitting methods for the nonlinear Dirac equation in the nonrelativistic limit regime, where solutions oscillate with $O(\varepsilon^2)$ time wavelength. Surprisingly, both methods exhibit super-resolution, accurately capturing solutions even when the time step $\tau$ exceeds the $O(\varepsilon^2)$ wavelength, with uniform $1/2$-order convergence in $\varepsilon$, and improved $O(\tau)$ and $O(\tau^{3/2})$ convergence under non-resonant $\tau$, respectively.

ABSTRACT

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 limit regime with a small parameter $0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this limit regime, the solution highly oscillates in time with wavelength at $O(\varepsilon^2)$ in time. The splitting methods surprisingly show super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solution accurately even if the time step size $ au$ is much larger than the sampled wavelength at $O(\varepsilon^2)$. Similar to the linear case, $S_1$ and $S_2$ both exhibit $1/2$ order convergence uniformly with respect to $\varepsilon$. Moreover, if $ au$ is non-resonant, i.e. $ au$ is away from certain region determined by $\varepsilon$, $S_1$ would yield an improved uniform first order $O( au)$ error bound, while $S_2$ would give improved uniform $3/2$ order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.

Motivation & Objective

• To analyze the uniform error behavior of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic limit regime.
• To investigate whether splitting methods can achieve super-resolution, breaking the classical sampling limit under Shannon's theorem.
• To establish uniform error bounds independent of the small parameter $\varepsilon$, which is inversely proportional to the speed of light.
• To determine conditions under which improved convergence rates are achieved, particularly for non-resonant time steps.

Proposed method

• The analysis focuses on the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) for the nonlinear Dirac equation without external magnetic potentials.
• The methods decompose the equation into linear and nonlinear subproblems, solving them sequentially or symmetrically over each time step.
• A rigorous error analysis is conducted in the nonrelativistic limit regime with $0 < \varepsilon \leq 1$, where solutions exhibit $O(\varepsilon^2)$-oscillatory behavior in time.
• The study examines the impact of time step $\tau$ on convergence, particularly distinguishing between resonant and non-resonant $\tau$ relative to $\varepsilon$.
• Theoretical bounds are derived using asymptotic expansions and energy estimates to establish uniform convergence with respect to $\varepsilon$.
• Numerical experiments are performed to validate the theoretical findings on convergence rates and super-resolution behavior.

Experimental results

Research questions

• RQ1 Can Lie-Trotter and Strang splitting methods achieve uniform error bounds in the nonrelativistic limit regime of the nonlinear Dirac equation?
• RQ2 Do these splitting methods exhibit super-resolution, allowing accurate resolution of $O(\varepsilon^2)$-oscillatory solutions with time steps $\tau \gg \varepsilon^2$?
• RQ3 What is the convergence rate of $S_1$ and $S_2$ when $\tau$ is non-resonant, and how does it compare to the resonant case?
• RQ4 Is super-resolution preserved for higher-order splitting methods beyond $S_1$ and $S_2$?
• RQ5 How do the error bounds depend on the small parameter $\varepsilon$, and are they uniform across $\varepsilon \in (0,1]$?

Key findings

• Both Lie-Trotter ($S_1$) and Strang ($S_2$) splitting methods achieve uniform $1/2$-order convergence in time step $\tau$ with respect to the small parameter $\varepsilon$.
• When the time step $\tau$ is non-resonant, $S_1$ achieves an improved uniform first-order convergence rate of $O(\tau)$.
• Under non-resonant $\tau$, $S_2$ exhibits improved uniform $3/2$-order convergence, i.e. $O(\tau^{3/2})$.
• The methods demonstrate super-resolution, accurately resolving $O(\varepsilon^2)$-oscillatory solutions even when $\tau \gg \varepsilon^2$, violating the classical Shannon sampling limit.
• The super-resolution phenomenon is shown to persist for higher-order splitting methods beyond $S_1$ and $S_2$.
• Numerical results confirm the theoretical error bounds and the validity of super-resolution across various $\varepsilon$ and $\tau$ regimes.