A Class of Arbitrarily High‐Order Energy‐Conserving Schemes for the Nonlinear Schrödinger Wave Equation

ABSTRACT This article proposes a class of arbitrarily high‐order energy‐conserving schemes for the nonlinear Schrödinger wave equation, aiming to address the issue that existing conservative schemes generally either fail to preserve the original energy conservation property or suffer from low temporal accuracy. The compact difference method is employed for spatial discretization and the Petrov–Galerkin (PG) method for temporal discretization, thereby constructing a fully discrete numerical scheme. At the theoretical level, the discrete energy conservation law of the scheme is rigorously proven. Furthermore, an efficient linearized point‐wise fixed‐point iteration algorithm is developed to achieve fast solving of the derived implicit schemes. Numerically, a series of numerical experiments is carried out to verify the theoretical results and demonstrate that the proposed schemes outperform the existing temporal high‐order original energy‐conserving schemes in computational efficiency.