In this work, four numerical time‐splitting methods are proposed for the (1 + 1)‐dimensional nonlinear Dirac equation. All of these methods (or schemes) are proved to satisfy the charge conservation in the discrete level. To enhance the computation efficiency, the block Thomas algorithm is adopted. Numerical experiments are given to test the accuracy order for these schemes, to simulate numerically the binary collision including two standing waves and two moving solitons, meanwhile, the dynamic properties for the nonlinear Dirac equation are discussed. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1582–1602, 2017

In this work, a second-order accuracy in both space and time Crank–Nicolson (C-N)-type scheme, a fourth-order accuracy in space and second-order accuracy in time compact scheme and a sixth-order accuracy in space and second-order accuracy in time compact scheme are proposed for the derivative nonlinear Schrödinger equation. The C-N-type scheme is tested to satisfy the conservation of discrete mass. For the two compact schemes, the iterative algorithm and the Thomas algorithm in block matrix form are adopted to enhance the computational efficiency. Numerical experiment is given to test the mass conservation for the C-N-type scheme as well as the accuracy order of the three schemes. In addition, the numerical simulation of binary collision and the influence on the solitary solution by adding a small random perturbation to the initial condition are also discussed.