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

Abstract

A high-order compact modified Crank-Nicolson scheme is proposed for the nonlinear Sobolev equation, in which a convex energy splitting approach is employed to address key computational challenges. By decomposing the energy functional 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 boundedness 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 experiments confirm that the expected convergence 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 numerical framework for complex nonlinear Sobolev-type problems.