The Conservative Numerical Scheme for the Hirota Equation

In this paper, we derive a semi-discrete scheme using the central difference method, which perfectly preserves the conservation of mass and energy for the Hirota equation. By applying the Crank–Nicolson method for temporal discretization, we develop the fully discrete scheme that conserves mass and energy. It is shown that the accuracy of the fully discrete scheme is of the second order in space and time. Because the Crank–Nicolson discretization leads to a nonlinear algebraic system, an efficient iterative solver is proposed that linearizes and solves the resulting five-diagonal matrix at each iteration while treating high-order contributions iteratively to reduce computational cost. Numerical experiments are presented to demonstrate the accuracy and verify the conservation properties.