Maximum Convergence of a Novel High‐Precision Conservative Compact Difference Scheme for the Coupled Nonlinear Schrödinger (CNLS) Equations

ABSTRACT In this work, a novel high‐order compact conservative difference scheme for the coupled nonlinear Schrödinger (CNLS) equations is researched. The scheme is three‐level nonlinear‐implicit. The biggest novelty of the proposed scheme is that it adopts the averaging technique based on the classic Crank–Nicolson method, and it has discrete mass conservation and energy conservation. The numerical analysis of the proposed scheme is discussed by using the standard discrete energy method, the matrix energy technique, and the mathematical induction. Theoretical results show that the numerical solution converges unconditionally, and the optimal convergent rate is proved to be at in the discrete maximum norm with grid size and time step . Finally, numerical experiments verify the correctness of the theoretical results and demonstrate the efficiency and accuracy of the proposed scheme.