A Second Order Energy Stable Scheme for the Cahn-Hilliard-Hele-Shaw Equations
This work provides a more accurate and stable numerical method for simulating phase-field models with convection, benefiting researchers in computational fluid dynamics and materials science.
The authors developed a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations that is uniquely solvable, unconditionally energy stable, and efficiently solvable with a nonlinear multigrid solver. They proved optimal convergence rates via an ℓ∞(0,T; H_h^1) error estimate and demonstrated accuracy and efficiency through numerical simulations.
We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an $\ell^2 (0,T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $\nabla \cdot (ϕ\boldsymbol{u})$, we perform an $\ell^\infty (0,T; H_h^1)$ error estimate instead of the classical $\ell^\infty (0,T; \ell^2)$ one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.