A linearly implicit local energy-dissipative method for the generalized nonlinear wave equation

In this paper, we propose a linearly implicit scheme that preserves the local energy dissipation property for the generalized nonlinear wave equation. By employing the energy quadratization approach, we reformulated the original equation into an equivalent form. A semi-discrete structure-preserving system was constructed via finite difference discretization in space. Subsequently, we derived a fully discretized scheme using the Crank-Nicolson method combined with the extrapolation technique, ensuring the preservation of local energy dissipation law. Under appropriate boundary conditions, such as homogeneous Dirichlet or periodic boundary conditions, the proposed method also maintained the global energy dissipation law. Furthermore, the unique solvability, fast implementation, and convergence theorem of the discrete scheme were analyzed rigorously. Numerical experiments are presented to validate our theoretical results, demonstrating that the proposed scheme outperforms traditional methods in terms of stability and efficiency.