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A two-cell Cahn–Hilliard mass and energy ledger

Example

Take two unit cells on $(0, 2)$ , one interior face with transmissibility $T = 1$ , no-flux boundary faces, $ε = κ = M = τ = 1$ , and $u^{0} = (1, - 1)$ . The discrete mean is zero, and the one-step convex-splitting state is $u^{1} = (a, - a)$ where $a$ is the unique root in $(0, 1)$ of $2 a^{3} + 7 a - 5 = 0$ . The initial energy is $2$ and the new energy is $2 a^{2} + \frac{1}{2} (1 - a^{2})^{2} < 2$ .

Facts & Assumptions

Given: The two-cell finite-volume realization and the convex-splitting Cahn–Hilliard step.

[F1]

No-flux finite-volume and conforming realizations for phase fields: for this two-cell mesh, $m_{h} (u, v) = u_{1} v_{1} + u_{2} v_{2}$ , $A_{h} (u_{1}, u_{2}) = (u_{1} - u_{2}, u_{2} - u_{1})$ , and $E_{h} (u) = \frac{1}{2} ⟨ A_{h} u, u ⟩_{h} + F (u_{1}) + F (u_{2})$ .

[F2]

Existence, uniqueness, mass conservation, and energy decay of a convex-splitting step: the Cahn–Hilliard step is unique, preserves the discrete mean, and does not increase $E_{h}$ .

Verification

1.1F1F2givenalgebra

The initial mean is zero, so mass conservation [F2] forces $u^{1} = (a, - a)$ . Since $A_{h} (a, - a) = (2 a, - 2 a)$ and $F_{c}^{'} (s) = s^{3} + s$ , $F_{e}^{'} (s) = 2 s$ , the first chemical-potential component is $μ_{1} = 2 a + (a^{3} + a) - 2 = a^{3} + 3 a - 2$ . The first state equation gives $a - 1 = - 2 μ_{1}$ , hence $2 a^{3} + 7 a - 5 = 0$ . The polynomial is strictly increasing because its derivative is $6 a^{2} + 7 > 0$ , and its values at $0$ and $1$ are $- 5$ and $4$ ; therefore it has exactly one root $a \in (0, 1)$ .

2.1F1step 1.1algebra∎

The initial gradient energy is $\frac{1}{2} ⟨ A_{h} u^{0}, u^{0} ⟩_{h} = 2$ and both potential terms vanish, so $E_{h} (u^{0}) = 2$ . For $u^{1} = (a, - a)$ the gradient energy is $2 a^{2}$ and the potential energy is $\frac{1}{2} (1 - a^{2})^{2}$ . Since $0 < a < 1$ , $(1 - a^{2})^{2} < 1 - a^{2}$ , hence $E_{h} (u^{1}) < 2 = E_{h} (u^{0})$ . This is the exact finite ledger for the step.

Depends on

Used by

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Sources

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, Vol. VII, 2000; complete updated author text (standard reference, not scraped)
Jing Guo, Cheng Wang, Steven M. Wise, and Xingye Yue, An H2 convergence of a second-order convex-splitting finite difference scheme for the three-dimensional Cahn–Hilliard equation, Communications in Mathematical Sciences 14 (2016), 489–515 (standard reference, not scraped)