A Second Order Positivity-Preserving Scheme for the Nonlocal Cahn-Hilliard System with Variable Mobility and Logarithmic Flory-Huggins Potential

RESUMO

A second order finite difference scheme is analyzed for the nonlocal Cahn-Hilliard system with variable mobility and the singular Flory-Huggins logarithmic potential. The temporal discretization employs a modified Crank-Nicolson formulation, while the mobility is treated explicitly to guarantee ellipticity and reduce computational cost. To strictly enforce the positivity-preserving property at the discrete level, a nonlinear regularization term is added into the numerical scheme. Rigorous analysis shows that the scheme is uniquely solvable and stable under a modified numerical energy. The well-known analytical challenge associated with the variable mobility has to be handled carefully. To overcome this, a convergence framework is constructed based on two non-standard techniques: (1) a higher-order asymptotic expansion (extending up to the third order in time and fourth order in space) to retain a sufficiently order of accuracy; (2) a two-stage error strategy, wherein a rough error estimate first guarantees the discrete boundedness of the mobility and nonlinear terms, followed by a refined error analysis that yields the optimal convergence rate. Numerical experiments validate the theoretical results and demonstrate the robustness of the proposed scheme.