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A new linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrödinger–Hirota equation. Optimal, second order convergence in the discrete \(H^1\)-norm is proved, assuming that \(\tau \), h and \(\tfrac{\tau ^4}{h}\) are sufficiently small, where \(\tau \) is the time-step and h is the space mesh-size. The convergence analysis is based on the investigation of a modified version of the proposed finite difference method, which is innovative and handles the stability difficulties due to the presence of a nonlinear derivative term in the equation. The efficiency of the proposed finite difference method is verified by results from numerical experiments.

A Linear Implicit Finite Difference Discretization of the Schrödinger–Hirota Equation