Analysis of an Ultra-Weak Discontinuous Galerkin Method with Generalized Numerical Fluxes for Time-Fractional Cahn-Hilliard Equation

Abstract

In this paper, we study the time-fractional Cahn-Hilliard (TFCH) equation, which models phase separation processes along with nonlocal memory effects. A fully-discrete numerical scheme is developed using the ultra-weak discontinuous Galerkin (UWDG) method in space with generalized numerical fluxes, coupled with a non-uniform L1 time-stepping scheme. The proposed method is demonstrated to preserve key physical properties of the continuous model, including mass conservation and energy dissipation. We establish the unconditional unique solvability of the fully-discrete scheme using the convex-concave splitting approach and derive stability bounds for the order parameter. An optimal a priori error estimate in the $L^2$-norm is established, confirming the convergence of the proposed scheme. Finally, numerical experiments are performed to validate the theoretically predicted convergence order for different choices of numerical fluxes. Numerical simulations confirm the scheme’s accuracy, demonstrating energy dissipation, mass conservation, and phase separation phenomena.

Keywords

Time-fractional Cahn-Hilliard (TFCH) equation / Nonuniform L1 scheme / Ultra-weak discontinuous Galerkin (UWDG) method / Optimal error estimate / 65M60 / 65M12 / 65M15 / 35K55 / 35R11

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Shanghai University