# 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:

- 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
- 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}

}