On numerical methods for the semi-nonrelativistic system of the nonlinear Dirac equation

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of $\mathcal{O}(\varepsilon^{-2})$, where $0 < ε \ll 1$ is inversely proportional to the speed of light. It was shown in [7], however, that such solutions can be approximated up to an error of $\mathcal{O}(\varepsilon^{-2})$ by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the exponential explicit midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy by a factor of six.