A Three‐Layer Finite Difference Scheme for the Nonlinear Kuramoto‐Tsuzuki Complex Equation With Variable Coefficients

ABSTRACT In this paper, we investigate a linearized three‐layer finite difference scheme for solving the variable‐coefficient nonlinear Kuramoto‐Tsuzuki (K‐T) complex equation. The proposed scheme employs the standard central finite difference method for spatial discretization, while a combination of the central averaging method and the central difference method is applied in the temporal direction. The nonlinear term is treated using a semi‐implicit linearization approach, which effectively enhances stability and reduces computational cost. We provide a rigorous analysis to establish the second‐order accuracy of the scheme, as well as the boundedness and uniqueness of the solution are proved by utilizing an energy‐based analytical method in conjunction with mathematical induction. Finally, numerical experiments are performed to validate the theoretical analysis.