TY - JOUR
T1 - Symmetric High Order Gautschi-Type Exponential Wave Integrators Pseudospectral Method for the Nonlinear Klein-Gordon Equation in the Nonrelativistic Limit Regime
AU - Wang , Yan
AU - Zhao , Xiaofei
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 405
EP - 427
PY - 2018
DA - 2018/03
SN - 15
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/12523.html
KW - Nonlinear Klein-Gordon equation, nonrelativistic limit, exponential wave integrator, high order accuracy, time symmetry, error estimate, meshing strategy, spectral method.
AB - <p style="text-align: justify;">A group of high order Gautschi-type exponential wave integrators (EWIs) Fourier
pseudospectral method are proposed and analyzed for solving the nonlinear Klein-Gordon equation
(KGE) in the nonrelativistic limit regime, where a parameter $0 &lt; ε\ll 1$ which is inversely
proportional to the speed of light, makes the solution propagate waves with wavelength $O(ε^2)$ in time and $O(1)$ in space. With the Fourier pseudospectral method to discretize the KGE in
space, we propose a group of EWIs with designed Gautschi’s type quadratures for the temporal
integrations, which can offer any intended even order of accuracy provided that the solution is
smooth enough, while all the current existing EWIs offer at most second order accuracy. The
scheme is explicit, time symmetric and rigorous error estimates show the meshing strategy of
the proposed method is time step $\tau = O(ε^2)$ and mesh size $h = O(1)$ as $0 &lt; ε \ll 1$, which is
‘optimal’ among all classical numerical methods towards solving the KGE directly in the limit
regime, and which also distinguish our methods from other high order approaches such as Runge-Kutta
methods which require $\tau = O(ε^3)$. Numerical experiments with comparisons are done to
confirm the error bound and show the superiority of the proposed methods over existing classical
numerical methods.</p>