An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations

In this work, an energy-preserving scheme is proposed for the nonlinear Dirac equation by combining the exponential time differencing method with the scalar auxiliary variable (SAV) approach. First, the original equations can be transformed into the equivalent systems by utilizing the SAV technique. Then the exponential time integrator method is applied for discretizing the temporal derivative, and the standard central difference scheme is used for approximating the spatial derivative for the equivalent one. Finally, the reformulated systems, the semi-discrete spatial scheme, and the fully-discrete, linearly implicit exponential scheme are proven to be energy conserving. The numerical experiments confirm the theoretical results.