A conservative higher-order finite difference scheme for solving the Gardner equation with dual power-law nonlinearities in both 1D and 2D

In this study, we have developed a conservative finite difference scheme that demonstrates fourth-order accuracy in space while preserving the discrete conservation of the invariant Mⁿ and energy Eⁿ for the Gardner equation with dual power-law nonlinearities in both 1D and 2D. Theoretical analyses and numerical results indicate that the proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space. The solvability, stability and convergence of the difference scheme are rigorously analyzed using the discrete energy method. Additionally, experimental results are provided to illustrate the effectiveness of the proposed difference scheme in accurately capturing the dynamics of the Gardner equation in both 1D and 2D, as well as its reliability for long-term simulations.