Fourth-Order Compact Finite Difference Method for the Schrödinger Equation with Anti-Cubic Nonlinearity

In this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserves both mass and energy through provable conservation properties. Under suitable assumptions on the exact solution, we establish upper and lower bounds for the numerical solution in the infinity norm, and further prove that the errors are fourth-order accurate in space and second-order in time in both the 2-norm and infinity norm. A detailed description of the nonlinear system solver at each time step is provided. We validate the proposed method through numerical experiments that demonstrate its efficiency, including fourth-order convergence (when sufficiently small time steps are used) and machine-level accuracy in the relative errors of mass and energy.