English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We propose and analyze a multiscale time integrator Fourier pseudospectral (MTIFP) method for solving the Klein-Gordon (KG) equation with a dimensionless parameter 0 < ε ≤ 1 which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. 0 < ε ≪ 1, the solution to the KG equation propagates waves with amplitude at O(1) and wavelength at O(ε) in time and O(1) in space, which causes significantly numerical burdens due to the high oscillation in time. The MTI-FP method is designed by adapting a multiscale decomposition by frequency (MDF) to the solution at each time step and applying an exponential wave integrator to the nonlinear Schrödinger equation with wave operator under well-prepared initial data for ε-frequency and O(1)amplitude waves and a KG-type equation with small initial data for the reminder waves in the MDF. We rigorously establish two independent error bounds in H-norm to the MTI-FP method at O(hm0 + τ + ε) and O(hm0 + τ/ε) with h mesh size, τ time step and m0 ≥ 2 an integer depending on the regularity of the solution, which immediately imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O(τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O(τ) in the regimes when either ε = O(1) or 0 < ε ≤ τ . Numerical results are reported to confirm the error bounds and demonstrate the efficiency and accuracy of the MTI-FP method for the KG equation, especially in the nonrelativistic limit regime.

A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime