Paper Info
Title
Uniformly Accurate Nested Picard Iterative Integrators for the Klein-Gordon Equation in the Nonrelativistic Regime
Abstract
In this paper, a class of uniformly accurate nested Picard iterative integrator (NPI) Fourier pseudospectral methods is proposed for the nonlinear Klein-Gordon equation (NLKG) in the nonrelativistic regime, involving a dimensionless parameter $$\varepsilon \ll 1$$ inversely proportional to the speed of light. For $$0<\varepsilon \ll 1$$ , the solution propagates waves in time with $$O(\varepsilon ^2)$$ wavelength, which brings significant difficulty in designing accurate and efficient numerical schemes. The idea of NPI methods can be applied to derive arbitrary higher-order methods in time with optimal and uniform accuracy (w.r.t. $$\varepsilon \in (0,1]$$ ). Detailed constructions of the NPI methods up to the third order in time are presented for NLKG with a cubic/quadratic nonlinear term, where the corresponding error estimates are rigorously analyzed. In addition, the practical implementation of the second-order NPI method via Fourier pseupospectral discretization is clearly demonstrated. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the proposed schemes.
Year
DOI
Venue
2022
10.1007/s10915-022-01909-5
Journal of Scientific Computing
Keywords
DocType
Volume
Nonlinear Klein-Gordon equation, Nonrelativistic limit, Picard iteration, Uniform convergence, Primary 35Q41, 65M70, 65N35
Journal
92
Issue
ISSN
Citations 
2
0885-7474
0
PageRank 
References 
Authors
0.34
12
2
Name
Order
Citations
PageRank
Yongyong Cai18011.43
Xuanxuan Zhou200.34