Treffer: A conservative fourth-order compact linearized difference scheme for the generalized Gardner equation with variable coefficients.

This article develops a high-order compact and linearized conservative difference scheme specifically designed to solve the Gardner equation with variable coefficients. We begin by establishing the law of energy conservation and providing a priori estimates for the Gardner equation. Next, we introduce a high-order compact and linearized conservative finite difference scheme for numerical simulation, which achieves second-order accuracy in time and fourth-order accuracy in space. Subsequently, we rigorously validate the energy conservation, solvability, convergence and stability of the numerical solution. Finally, we present several numerical examples to demonstrate the effectiveness and reliability of the conservative scheme for long-term simulations. [ABSTRACT FROM AUTHOR]