Unconditionally Convergent High-Order Energy-Stable Compact Scheme for Nonlinear Sobolev Equation through Convex Splitting

Source abstract

A high-order compact modified Crank-Nicolson scheme is proposed for the nonlinear Sobolev equation, in which a convex energy splitting approach is em ployed to address key computational challenges. By decomposing the energy func tional into two convex components, a spatially fourth-order and temporally second order compact finite difference discretization that naturally preserves the physical energy structure is developed. Rigorous theoretical analysis establishes the bound edness and unique solvability of the numerical solution and proves a discrete energy dissipation law. Furthermore, the scheme is proven to be unconditionally convergent (without any time-space mesh-ratio restriction) in the discrete norm. Numerical ex periments confirm that theexpectedconvergence rates are achieved, and the scheme displays long-term stability and consistent energy decay. These results suggest that the proposed method provides an accurate, efficient, and structure-preserving nu merical framework for complex nonlinear Sobolev-type problems.