Author:
Jahnke Tobias,Kirn Michael
Abstract
AbstractSolving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of $${\mathscr {O}} \! \left( \varepsilon ^{-2}\right) $$
O
ε
-
2
, where $$0<\varepsilon \ll 1$$
0
<
ε
≪
1
is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of $${\mathscr {O}} \! \left( \varepsilon ^2\right) $$
O
ε
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 explicit exponential 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 significantly.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Computer Networks and Communications,Software