Three high-order structure-preserving schemes for the generalized Rosenau-RLW-Burgers equation

This study introduces three high-order difference schemes for solving the generalized Rosenau-RLW-Burgers equation with a power-law nonlinearity $ u^{p}u_{x} $ upux ( $ p\geq 1 $ p≥1). By employing implicit high-order operators, the proposed schemes preserve the fundamental structural properties of the continuous equation. Through rigorous mathematical analysis, these schemes are shown to satisfy discrete mass conservation and energy dissipation laws, and achieve optimal convergence rates of $ O(\tau ^{2}+h^{4}) $ O(τ2+h4), $ O(\tau ^{2}+h^{6}) $ O(τ2+h6) and $ O(\tau ^{2}+h^{8}) $ O(τ2+h8) in both the $ \|\cdot \| $ ‖⋅‖-norm and the maximum norm $ \|\cdot \|_{\infty } $ ‖⋅‖∞. Combining spectral matrix decomposition with discrete energy analysis, we rigorously establish the existence, uniqueness, numerical stability and convergence of the numerical solutions. Extensive numerical experiments confirm the accuracy and efficiency of the proposed schemes. Comparative results demonstrate that the modified linearized scheme offers superior performance in balancing computational efficiency and numerical accuracy among the three high-order schemes.