Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime

Abstract

We propose two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral (SPEWIFP) methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. In this regime, the solution of Dirac equation is highly oscillatory in time because of the small parameter $0 <\varepsilon \ll 1$ which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition $\tau \lesssim 1$ and preserve the modified energy and modified mass in the discrete level. Although our methods can only preserve the modified energy and modified mass instead of the original energy and mass, our methods are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are often implicit. Through rigorous error analysis, we give the error bounds of the methods at $O(h^{m_0} + \tau ^2/\varepsilon ^2)$ where h is mesh size, $\tau$ is time step and the integer $m_0$ is determined by the regularity conditions. These error bounds indicate that, to obtain the correct numerical solution in the simultaneously massless and nonrelativistic regime, our methods request the $\varepsilon$-scalability as $h = O(1)$ and $\tau = O(\varepsilon )$ which is better than the $\varepsilon$-scalability of the finite difference (FD) methods: $h =O(\varepsilon ^{1/2})$ and $\tau = O(\varepsilon ^{3/2})$. Numerical experiments confirm that the theoretical results in this paper are correct.