Pseudospectral methods with PML for nonlinear Klein-Gordon equations in classical and non-relativistic regimes

Abstract

Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation are proposed and analyzed. To obtain a high spatial accuracy, new regularized Bermúdez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme should be preferred for accuracy purpose when used within the framework of pseudo-spectral methods combined with the regularized Bermúdez type functions. In addition, in the non-relativistic regime, numerical examples lead to the conclusion that the error related to regularized Bermúdez type absorption functions is insensitive to the small parameter $\epsilon$ involved in the NKGE. The paper ends by a two-dimensional example showing that the strategy extends to the rotating NKGE where the vortex dynamics is very well-reproduced.

26 pages

References in corpus (4)

• Notes on Perfectly Matched Layers (PMLs)
• Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
• Long time error analysis of finite difference time domain methods for the nonlinear Klein-Gordon equation with weak nonlinearity
• Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation