skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations

Abstract

In this paper, we consider the multi-symplectic Runge-Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK methods. Numerical experiments presented reveal that MSRK methods applied to the nonlinear Dirac equation preserve exactly conservation laws of charge and momentum, and conserve the energy conservation in the corresponding numerical accuracy to the method utilized. It is verified numerically that MSRK methods are stable and convergent with respect to the conservation laws of energy, momentum and charge, and MSRK methods preserve not only the inner geometric structure of the equation, but also some crucial conservative properties in quantum physics. A remarkable advantage of MSRK methods applied to the nonlinear Dirac equation is the precise preservation of charge conservation law.

Authors:
 [1];  [2]
  1. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080 (China). E-mail: hjl@lsec.cc.ac.cn
  2. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080 (China) and Graduate School of the Chinese Academy of Sciences, Beijing 100080 (China). E-mail: lichun@lsec.cc.ac.cn
Publication Date:
OSTI Identifier:
20687273
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 211; Journal Issue: 2; Other Information: DOI: 10.1016/j.jcp.2005.06.001; PII: S0021-9991(05)00282-2; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; CHARGE CONSERVATION; COMPARATIVE EVALUATIONS; CONSERVATION LAWS; DIRAC EQUATION; ENERGY CONSERVATION; NONLINEAR PROBLEMS; QUANTUM MECHANICS; RELATIVISTIC RANGE; RUNGE-KUTTA METHOD

Citation Formats

Hong Jialin, and Li Chun. Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. United States: N. p., 2006. Web. doi:10.1016/j.jcp.2005.06.001.
Hong Jialin, & Li Chun. Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. United States. doi:10.1016/j.jcp.2005.06.001.
Hong Jialin, and Li Chun. Fri . "Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations". United States. doi:10.1016/j.jcp.2005.06.001.
@article{osti_20687273,
title = {Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations},
author = {Hong Jialin and Li Chun},
abstractNote = {In this paper, we consider the multi-symplectic Runge-Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK methods. Numerical experiments presented reveal that MSRK methods applied to the nonlinear Dirac equation preserve exactly conservation laws of charge and momentum, and conserve the energy conservation in the corresponding numerical accuracy to the method utilized. It is verified numerically that MSRK methods are stable and convergent with respect to the conservation laws of energy, momentum and charge, and MSRK methods preserve not only the inner geometric structure of the equation, but also some crucial conservative properties in quantum physics. A remarkable advantage of MSRK methods applied to the nonlinear Dirac equation is the precise preservation of charge conservation law.},
doi = {10.1016/j.jcp.2005.06.001},
journal = {Journal of Computational Physics},
number = 2,
volume = 211,
place = {United States},
year = {Fri Jan 20 00:00:00 EST 2006},
month = {Fri Jan 20 00:00:00 EST 2006}
}
  • We are presenting a family of trigonometrically fitted partitioned Runge-Kutta symplectic methods of fourth order with six stages. The solution of the one dimensional time independent Schroedinger equation is considered by trigonometrically fitted symplectic integrators. The Schroedinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator and the exponential potential.
  • In this work we consider explicit Symplectic Partitioned Runge-Kutta methods (SPRK) with five stages for problems with separable Hamiltonian. We construct a new method with constant coefficients third algebraic order and eighth phase-lag order.
  • The temporal integration of hyperbolic partial differential equations (PDEs) has been shown to lead sometimes to the deterioration of accuracy of the solution because of boundary conditions. A procedure for removal of this error in the linear case has been established previously. In this paper the authors consider hyperbolic PDEs (linear and nonlinear) whose boundary treatment is accomplished via the simultaneous approximation term (SAT) procedure. A methodology is presented for recovery of the full order of accuracy and has been applied to the case of a fourth-order explicit finite-difference scheme.
  • We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the confolution kernal via a discrete operational calculus. 23 refs., 2 tabs.
  • Generalized Runge-Kutta methods specifically devised for the numerical solution of stiff systems of ordinary differential equations are described. An A-stable method is employed to solve several sample point reactor kinetics problems, explicitly showing the quantities required by the method. The accuracy and speed of calculation as obtained by implementing the method in a microcomputer are found to be acceptable.