In this work, a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrödinger equation. We verify the mass conservation for the two-level implicit scheme. The influence on the soliton solution by adding a small random perturbation to the initial condition is discussed. The numerical experiments are given to test the accuracy order for different schemes, respectively. We also test the conservative property of mass and Hamiltonian for these schemes from the numerical point of view.

Based on the Lie-algebra, a new time-compact scheme is proposed to solve the one-dimensional Dirac equation. This time-compact scheme is proved to satisfy the conservation of discrete charge and is unconditionally stable. The time-compact scheme is of fourth-order accuracy in time and spectral order accuracy in space. Numerical examples are given to test our results.

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