High‐Order Conservative Crank–Nicolson Difference Scheme for the Hirota–Satsuma Coupled KdV Equation

ABSTRACT This paper introduces a high‐order, conservative Crank–Nicolson finite difference scheme for the Hirota–Satsuma coupled KdV (HS‐KdV) equation, achieving fourth‐order accuracy in space and second‐order accuracy in time. Unlike existing low‐order, non‐conservative, or computationally expensive methods, the nonlinear system is solved via fixed‐point iteration at each time step, exactly preserving discrete mass and energy. We rigorously prove solvability, conservation, stability, and convergence with order in the discrete norm. By employing a novel cut‐off function technique combined with mathematical induction, we derive optimal ‐norm error estimates without requiring a priori boundedness of the numerical solution. Extensive numerical experiments involving single, double, and triple solitary wave interactions confirm the theoretical convergence rates and exact preservation of invariants, demonstrating the scheme's robustness and superior conservation properties in simulating long‐term, complex HS‐KdV wave dynamics.