Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

  • Authors:

  • Weizhu Bao;Xuanchun Dong

  • Affiliations:

  • National University of Singapore, Department of Mathematics and Center for Computational Science and Engineering, 119076, Singapore, Singapore;National University of Singapore, Department of Mathematics and Center for Computational Science and Engineering, 119076, Singapore, Singapore

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2012

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Abstract

We analyze rigourously error estimates and compare numerically temporal/spatial resolution of various numerical methods for solving the Klein–Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter $${0 O(1) in time and space, respectively. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter $${{\varepsilon}}$$. Based on the error bounds, in order to compute ‘correct’ solutions when $${0 τ = O(1) and $${\tau=O({\varepsilon}^2)}$$ for linear and nonlinear KG equations, respectively, when $${0