American Institute of Mathematical Sciences

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May  2006, 6(3): 623-640. doi: 10.3934/dcdsb.2006.6.623

Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model

Sihong Shao 1, and  Huazhong Tang 1,

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China, China

Received  June 2005 Revised  December 2005 Published  February 2006

This paper extends Runge-Kutta discontinuous Galerkin (RKDG) methods to a nonlinear Dirac (NLD) model in relativistic quantum physics, and investigates interaction dynamics of corresponding solitary wave solutions. Weak inelastic interaction in ternary collisions is first observed by using high-order accurate schemes on finer meshes. A long-lived oscillating state is formed with an approximate constant frequency in collisions of two standing waves; another is with an increasing frequency in collisions of two moving solitons. We also prove three continuum conservation laws of the NLD model and an entropy inequality, i.e. the total charge non-increasing, of the semi-discrete RKDG methods, which are demonstrated by various numerical examples.
Keywords: interaction dynamics., Runge-Kutta discontinuous Galerkin methods, nonlinear Dirac equations, the total charge non-increasing.
Mathematics Subject Classification: Primary: 65M60, 81Q05; Secondary: 35L6.
Citation: Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623
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