Finite-volume method for the Cahn-Hilliard equation with dynamic boundary conditions

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1 Finite-volume method for the Cahn-Hilliard equation with dynamic oundary conditions Flore Naet o cite this version: Flore Naet. Finite-volume method for the Cahn-Hilliard equation with dynamic oundary conditions. Congrés SAI 2013, ar 2014, Carry-le-Rouet, France. 45, Congrés SAI 2013, ESAI : Proceedings and Surveys. <hal v3> HAL Id: hal Sumitted on 18 ay 2014 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are pulished or not. he documents may come from teaching and research institutions in France or aroad, or from pulic or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, puliés ou non, émanant des étalissements d enseignement et de recherche français ou étrangers, des laoratoires pulics ou privés.

2 ESAI: PROCEEDINGS, Vol.?, 2014, 1-10 Editors: Will e set y the pulisher FINIE-VOLUE EHOD FOR HE CAHN-HILLIARD EQUAION WIH DYNAIC BOUNDARY CONDIIONS Flore Naet 1 Astract. A numerical scheme is proposed here to solve a diphasic Cahn-Hilliard equation with dynamic oundary conditions. A finite-volume method is implemented for the space discretization and existence and convergence results are proved. Numerical simulations are also presented that show the influence of these oundary conditions. 1. Introduction he Cahn-Hilliard equation descries the evolution of inary mixtures when, for example, a inary alloy is cooled down sufficiently. his prolem has een extensively studied for many years with Neumann oundary conditions. Recently, physicists [5 7] have introduced new oundary conditions, usually called dynamic oundary conditions, to account for the effective interaction etween the wall and the two mixture components in a confined system. With these dynamic oundary conditions, the Cahn-Hilliard equation results in the following system: Find the concentration c : [0, [ Ω R such that: t c = Γ µ, in 0, Ω; µ = εσ c + σ ε f c, in 0, Ω; c0,. = c 0, in Ω; ε 3 t c Γ = ε 2 σ s σ c Γ σ f Γ s Γ sc Γ εσ n c, on 0, Γ; n µ = 0, on 0, Γ; where µ is an intermediate unknown called chemical potential. here cannot e any mass exchange through the oundary, which is why we consider the homogeneous Neumann oundary condition for the chemical potential. he domain Ω R 2 is smooth, connected and ounded, with Γ = Ω its oundary and > 0 the final time. he Laplace-Beltrami operator on Γ is noted, n is the normal derivative at the oundary and c Γ is the trace of c on Γ. hese dynamic oundary conditions induced us to look for a solution in L 0,, H 1 Ω whose trace is in L 0,, H 1 Γ see heorem 4.5. he parameter ε > 0 accounts for the interface thickness, the coefficient Γ > 0 is the ulk moility and σ > 0 is the fluid-fluid surface tension. On the oundary, Γ s > 0 defines a surface kinetic coefficient and σ s > 0 a surface capillarity coefficient. he nonlinear terms f and f s represent respectively the ulk free energy density and the surface free energy density and they satisfy the following assumptions: 1 Aix-arseille Université, CNRS, Centrale arseille, LAP, UR 7353, 13453, arseille, France flore.naet@univ-amu.fr. c EDP Sciences, SAI 2014 P

3 2 ESAI: PROCEEDINGS Dissipativity: lim inf f x > 0 and lim inf f s x > 0. H diss x x Polynomial growth for f : there exist C > 0 and a real p 2 such that: f x C 1 + x p, f x C 1 + x p 1, f x C 1 + x p 2. H f 0 1 Figure 1. ypical choice for f : f c = c 2 1 c 2. Remark 1.1. We can notice that if we choose σ s = 0, Γ s = + and f s = 0, we formally recover the standard Neumann oundary condition n c = 0. he Cahn-Hilliard equation with dynamic oundary conditions P is such that the total free energy functional defined y ε Fc = Ω 2 σ c 2 + σ ε 2 ε f c + Γ 2 σ sσ c Γ 2 + σ f s c Γ, 1 is decreasing with respect to time: d dt Fct,. = Γ µt,. 2 ε3 Ω Γ s Γ Γ t c Γ t,. 2, t [0, [. From a mathematical point of view, prolem P has already een studied in [9 11] where questions such as gloal existence and uniqueness, existence of a gloal attractor, maximal regularity of solutions and convergence to an equilirium have een answered. From a numerical point of view, some numerical schemes have een considered in [5 7] in a finite-difference framework ut without proof of convergence. In [2], the authors propose a spatial finite-element semi-discretization and prove error estimate and convergence results on a sla with periodic conditions in the lateral directions and dynamic conditions in the vertical directions, so that complex geometries of the domain are not taken into account in the convergence analysis. In this paper, we investigate a finite-volume scheme for the space discretization of this prolem. his method is well adapted to the coupling of the dynamics in the domain and those on the oundary y the flux term n c. oreover, this kind of scheme preserves the mass and accounts naturally for the non-flat geometry of the oundary and for the associated Laplace-Beltrami operator. In Section 2, we recall the main finite-volume notation, for example used in [4], that we adapt to our prolem with a curved domain and dynamic oundary conditions. In Section 3, we give the discrete energy functional and the associated energy estimates. hen, we propose a finite-volume scheme with different time discretizations for the nonlinear terms. Existence and convergence results are stated in Section 4. Finally, we give some numerical results in Section 5 with different nonlinear terms on the oundary. 2. he discrete framework We give in this section the main notation and definitions used in this paper.

4 2.1. he discretization ESAI: PROCEEDINGS 3 Since Ω is a curved domain, the notation Fig. 2 and definitions are slightly different from the usual finitevolume definitions given for example in [4]. An admissile mesh is constituted of an interior mesh and a oundary mesh. he interior mesh is a set of control volumes we specify that some control volumes are curved Ω such that: if L, we have L = ; if L such that the dimension of L is equal to 1, then L is an edge of the mesh; = Ω. We note the set of edges of the control volumes in included in Γ we remark that these are not segments ut curved sections. We will use two different notations for an element of : we note e when we consider it as a control volume elonging to and we note σ when we consider it as the edge of an interior control volume. Let E e the set of the edges of the mesh, E ext = is the set of exterior edges and E int = E \ E ext is the set of interior edges. Let m σ e the length of the edge σ E. For each control volume, we associate a point x and we assume that for all neighouring control volumes, L the edge σ = L E int is orthogonal to the straight line going through x and x L. he distance etween x and x L is noted d,l and n L is the unit normal vector going from to L. We define y the polygon shaped y the vertices of if there exists at least an edge of on the oundary and, if not, =. Let m respectively m e the Leesgue measure of respectively. For any e, we note ẽ the chord associated with e, mẽ its length. We define x e as the intersection etween Γ and the straight line passing throught x and orthogonal to ẽ. Let y e e the intersection etween the line x x e and the chord ẽ. We note d,e the distance etween x and y e and n e is the unit normal vector to ẽ outward to. Let V e the set of the vertices included in Γ and d e,v e the distance etween the center y e and the vertex v V. For a vertex v = e e Γ which separates the control volumes e, e, d e,e is equal to the sum of d e,v and d e,v. We can notice that the proposed scheme uses only the coordinates of the vertices of the mesh in Γ and not the equation of the oundary Γ. y e x e n e y e d e,v v = e e n L d e,v x e Boundary mesh Interior mesh Vertex v V d,e x d,l x L Centers Figure 2. Finite-volume meshes he mesh size is defined y: size = sup{diam, }. All the constants in the results elow depend on a certain measure of the regularity of the mesh, which is classical and that we do not make explicit here in order to e more synthetic. In short, it is necessary that the control volumes do not ecome flat when the mesh is refined. Let N N and ]0, + [. he temporal interval [0, ] is uniformly discretized with a fixed time step t = N. For n {0,, N}, we define tn = n t.

5 4 ESAI: PROCEEDINGS 2.2. Discrete unknowns For a given time step t n, the FV method associates with all interior control volumes an unknown value c n and with all oundary control volumes e an unknown value c n e for the order parameter. he same notations are used for the chemical potential with an unknown value µ n for all. Because of the Neumann oundary condition, it is not necessary to have oundary unknows for µ. Whenever it is convenient, we associate with a discrete function u R the piecewise constant functions = u, u where u = u 1 L Ω and u = u e 1 e L Γ. u We note u t for all t [t n, t n+1 [: e respectively u t the piecewise constant function in ]0, [ Ω respectively ]0, [ Γ such that 2.3. Discrete inner products and norms u t t, x = u n+1 if x and u t t, x = u n+1 e if x e. Definition 2.1 Discrete L 2 norms. For u R, the L 2 Ω discrete norm of u is defined y: u 2 0, = m u 2. For u R, the L 2 Γ discrete norm of u is defined y: u 2 0, = e mẽu 2 e. Definition 2.2 Discrete H 1 semi-definite inner products. For u, v R, the H 1 Ω discrete semi-definite inner product is defined y: u, v 1, = u u L v v L m σ d,l + u u e mẽd,e σ E int d,l d,l σ E ext d,e v v e where, y convention, u e = u for σ = e E ext an edge of if u satisfies the homogeneous Neumann oundary condition. We note u 1, = u, u 1 2 1, the associated discrete H 1 Ω seminorm. For u, v R, the H 1 Γ discrete semi-definite inner product is defined y: u, v 1, = v=e e V d e,e ue u e ve v e d e,e d e,e We note u 1, = u, u 1 2 1, the associated discrete H 1 Γ seminorm. Now, we can define the H 1 discrete norms y: u 2 1, = u 2 0, + u 2 1,, u R and u 2 1, = u 2 0, + u 2 1,, u R Numerical scheme. 3. Numerical scheme and energy estimates We use a consistent two-point flux approximation for Laplace operators in Ω and a consistent two-point flux approximation for the Laplace-Beltrami operator on Γ. For nonlinear terms, we use two different discretizations descried elow, fully implicit and semi-implicit, so that we have to use a Newton method at each iteration. We assume that c n R is given. he scheme then writes as follows. d,e,

6 ESAI: PROCEEDINGS 5 Prolem 3.1. Find c n+1, µ n+1 R R such that u R, v R : ε 3 Γ Γ s e c n+1 m mẽ c n t v = Γ µ n+1, v 1, m µ n+1 u =εσ c n+1 e c n e t m σ σ E int d,l + εσ σ E ext d,e c n+1 c n+1 L u u L mẽ c n+1 c n+1 e = ε 2 σ σ s c n+1, u 1, σ εσ mẽ σ E ext d,e u + σ ε e c n+1 e c n+1 ue mẽd f s c n e, c n+1 e u e m d f c n, c n+1 u S he functions d f and d fs represent the discretizations for nonlinear terms f c and f sc. We can notice that in scheme S the coupling etween interior and oundary unknowns is performed y the two oxed terms: one in the interior mesh and the other on the oundary mesh. We can also remark that scheme S only uses geometric quantities related to the polygonal approximations of the control volumes. However, the convergence analysis is performed y using the exact geometric quantities related to the curved control volumes. In particular, we use projections of continuous functions on these curved control volumes which are useful to otain a suitale approximation for the initial data. In order to simplify the presentation and the analysis, we have written the scheme as a formulation which looks like a variational formulation. However, if for each control volume we choose the indicator function of this particular control volume as a test function in S, we recognize a usual finite-volume flux alance equation Energy estimate Here we give the definition of the discrete energy and the corresponding estimate. Definition 3.2 Discrete free energy. he discrete free energy associated with the continuous free energy 1 is composed of a ulk energy F, and a surface energy F s, such that for all c R : where: F, c = σ ε F c = F, c + F s, c m f c + ε 2 σ c 2 1, and F s, c = σ e mẽf s c e + ε2 2 σ σ s c 2 1,. By using Prolem 3.1 with v = µ n+1 and u = c n+1 c n as test functions, we otain the following energy estimate. Proposition 3.3 General energy estimate. Let c n R. We assume that there exists a solution c n+1, µ n+1 to Prolem 3.1. hen, the following equality holds: F c n+1 F c n + tγ µ n ε3 1 c n+1 1, c n Γ Γ s t + ε 2 σ c n+1 c n 2 + ε2 1, 2 σ σ s c n+1 c n 2 1, = σ ε + σ mẽ e 2 0, m f c n+1 f c n d f c n, c n+1 c n+1 c n fs c n+1 e f s c n e d f s c n e, c n+1 e c n+1 e c n e. 2

7 6 ESAI: PROCEEDINGS 3.3. Discretization for nonlinear terms In this section, we detail our two discretizations for the nonlinear terms used in scheme S and we give the associated energy estimates Fully implicit discretization For the fully implicit discretization in time, we choose d f and d f s independent of c n namely: d f c n, c n+1 = f c n+1, and d f s c n e, c n+1 e = f sc n+1 e, e. hen, y using the energy estimate 2 and dissipativity assumptions H diss, we otain the following discrete energy inequality: Proposition 3.4 Discrete energy inequality. Let c n R. We assume that there exists a solution c n+1, µ n+1 to Prolem 3.1. hen, there exists t 0 > 0 such that for all t t 0, we have: F c n+1 + tγ 2 + ε 4 σ µ n+1 c n ε3 1 1, Γ Γ s 2 t c n 2 1, c n+1 + ε2 2 σ σ s c n+1 c n 2 0, c n 2 F 1, c n. We can notice that t 0 depends on the parameters of the equation, so with this discretization we have to choose a small enough t. his is why we introduce elow another discretization Semi-implicit discretization We would like to otain an energy estimate without any condition on t. hus, we choose a discretization for nonlinear terms such that the right hand side in 2 is equal to 0: d f x, y = f y f x y x and d fs x, y = f sy f s x, x, y. y x We can remark that the potentials used for numerical tests are polynomials. hus, we can express d f respectively d f s as a polynomial in the variales x, y. We thus otain the following energy equality true for all t > 0: Proposition 3.5 Discrete energy equality. Let c n R. We assume that there exists a solution c n+1, µ n+1 to Prolem 3.1, then we have: F c n+1 + tγ µ n+1 + ε 2 σ c n+1 c n 2 + ε3 1 1, Γ Γ s t 2 1, c n+1 + ε2 2 σ σ s c n+1 c n 2 0, c n 2 = F 1, c n. We can notice that we use here two different discretizations for nonlinear potentials, ut we can choose another discretization such as, for example, the convex-concave discretization see [1] for more details. 4. Existence and convergence We give general assumptions on the discretization of nonlinear potential d f to demonstrate the existence and convergence theorems. d f is of C 1 class and there exist C 0 and a real p such that 2 p < +, d f a, C 1 + a p 1 + p 1 and D d f a,. C 1 + a p 2 + p 2. H d f

8 4.1. Existence ESAI: PROCEEDINGS 7 he existence of a solution to discrete Prolem 3.1 is ased on the topological degree theory and the a priori energy estimates otained aove. heorem 4.1 Existence of a discrete solution. Let c n R. Assuming that dissipativity assumptions H diss and growth conditions H d f hold and that there exist constants cn, cn s depending possily on c n such that, for all u R, m f u f c n d f c n, u u c n cn, e mẽ fs u e f s c n e d fs c n e, u e u e c n e cn s. hen, there exists at least one solution c n+1, µ n+1 R R to Prolem Convergence In order to prove the convergence result we have to define a solution to Prolem P in a weak sense. Definition 4.2 Weak formulation. We say that a couple c, µ L 0, ; H 1 Ω L 2 0, ; H 1 Ω such that rc L 0, ; H 1 Γ is solution to continuous Prolem P in the weak sense if for all ψ Cc [0, [ Ω, the following identities hold: 0 0 Ω Ω t ψc + Γ µ ψ = µψ + εσ c ψ + σ = ε3 rc 0 ψ0,.. Γ Γ s Γ Ω c 0 ψ0,., 4 ε f cψ + ε3 t ψc Γ + σ s σ ε 2 c Γ ψ + σ f 0 Γ Γ Γ sc Γ ψ s heorem 4.3 Bounds of the solutions. Assuming that assumptions H diss, H f, H d f, 3 hold and that there exists a constant C > 0 such that, for all n N, F c n+1 + C F c n, tγ µ n ε3 1 1, Γ Γ s t c n+1 c n 2 + ε 0, 2 σ then, there exists > 0 independent of and t such that: sup c n 1,, sup c n 1,, n N n N t N 1 t n=0 c n+1 2 c n t 1, N 1 and t n=0 c n+1 t N 1 n=0 c n+1 c n 2 1, + ε2 2 σ σ s c n+1 t µ n+1 2, 1, c n t c n hese ounds are one of the key elements to prove the convergence result elow y using the discrete H 1 compactness and the olmogorov theorem. We also note that we have nonlinearities in the domain Ω and on the oundary Γ. hus L 2 0, Ω compactness is not sufficient and we have to prove uniform estimates of time and space translates on Ω and Γ. 2 1,. 2 1, 3 5 6

9 8 ESAI: PROCEEDINGS heorem 4.4 Estimation of time and space translates. Let c t, µ t e a solution to Prolem 3.1, then there exists C > 0 not depending on size and t such that: c t. + τ,. c t.,. 2 Cτ, L 2 R R 2 c t. + τ,. c t.,. 2 Cτ, L 2 R Γ c t.,. + η c t.,. 2 C η, L 2 R R 2 c t., τ η. c t.,. 2 C η, L 2 R Γ where we note c t respectively c t the extension y 0 of c t respectively c t on R R 2 respectively R Γ and τ η represents the shifting of length η along the oundary Γ an orientation eing given on Γ. heorem 4.4 is proved y using the ounds of the solutions given y heorem 4.3 and scheme S. his heorem is essential to apply the olmogorov theorem and to prove the following convergence result. heorem 4.5 Convergence theorem. Assuming that conditions H diss, H f, H d f, 3 and 6 hold, let us consider Prolem P with an initial condition c 0 H 1 Ω such that rc 0 H 1 Γ. hen, there exists a weak solution c, µ on [0, [ in the sense of Definition 4.2. Furthermore, let c m, c m and Γ µ m m N m N e a sequence of solutions to Prolem 3.1 with a sequence of discretizations such that the space and time steps, h m and t m respectively, tend to 0. hen, up to a susequence, the following convergence properties hold, for all q 1, when h m, t m 0: c m c in L 2 0, ; L q Ω strongly, c m Γ rc in L 2 0, ; L q Γ strongly, and µ m µ in L 2 0, ; L q Ω weakly. Remark 4.6. We chose the initial concentration in the scheme equal to the mean-value projection: 1 c 0 = m c 0 1, c 0. m e e e 5. Numerical simulations In this section, we present numerical experiments for three different nonlinear surface free energy densities. We choose here the semi-implicit discretization in time for nonlinear terms in order to allow for a not too small time step t. For each simulation we consider the usual doule-well ulk potential f c = c 2 1 c 2. We are interested here y two domains with a Delaunay triangular mesh: a 0, 8 0, 4 rectangle with periodic oundary conditions in the lateral direction and dynamic oundary conditions in the vertical direction; a smooth curved domain whose diameter is equal to 2 and with dynamic oundary conditions everywhere on Γ. In these cases, x is the circumcenter of the control volume and y e is the middle of the chord ẽ. For each domain, we choose a random initial c 0 R etween 0.4 and 0.6 and we keep this initial data for each simulation with the same domain. We can then oserve the influence of the oundary conditions on the phase separation dynamics Influence of the surface diffusion term First, we choose f s = f for the surface potential and to compare the results in [5, 6], we egin with the rectangular domain Fig. 3 and the following parameters: ε = 0.3, Γ = σ = 0.1 for the ulk, Γ s = 10 for the

10 ESAI: PROCEEDINGS a σ s = 0 σ s = 5 Figure 3. Spinodal decomposition for rectangular domain surface and = 0.75, dt = 0.05 for the time. We oserve the influence of surface diffusion y computing the solution with two different values for the surface coefficient σ s. In oth case, we have lateral anisotropic structures ut their length scale is different. Indeed, when we have σ s = 0 Fig. 3a the structure length scales are shorter than when we have σ s = 5 Fig. 3. hese results are very close to those oserved in [5]. Now, we test the scheme with our curved domain Fig. 4 with the following parameters: ε = Γ = σ = 0.1 for the ulk, Γ s = 10 for the surface and = 0.025, dt = for the time a σ s = 0 σ s = 5 Figure 4. Spinodal decomposition for curved domain We oserve the same ehavior as for the rectangular domain Fig. 3: for σ s = 0 Fig. 4a, we have small typical structures on the oundary while for σ s = 5 Fig. 4a the structures are large excepted where the domain is too narrow Preferential attraction y the wall For the following computation Fig 5, we want to oserve the influence of the surface potential y taking f s c = g s c 2 h s +g s c where h s 0 descries the possile preferential attraction of one of the two components y the wall. hus, we choose fixed parameters: ε = 0.2, Γ = σ = 0.1 for the ulk, Γ s = 10, σ s = 0, g s = 10 for the surface and = 0.37, dt = for the time and we modify the coefficient h s. First, we notice than the parallel structures oserved when h s = 0 Fig. 5a are similar to those oserved in [2,7]. hen, we confirm the preferential attraction of the phase c = 1 y the oundary when h s > 0 Fig. 5 and we notice that this attraction changes all the ehavior in the domain Ω.

11 10 ESAI: PROCEEDINGS a h s = 0 h s = 3 Figure 5. Influence of h s Conclusion We propose here a finite-volume scheme to deal with the 2D Cahn-Hilliard equation with dynamic oundary conditions. With this method the coupling etween the equation in the domain and the equation on the oundary is easy to implement even with a curved geometry for the mesh. Furthermore, we give a convergence result which additionally enales to otain the existence of weak solutions for the continuous prolem. We can specify that we have error estimates for the Cahn-Hilliard equation with Neumann oundary conditions. oreover, in [8] we performed numerical simulations for error estimates which gave the expected first-order convergence. Finally, a possile future work will e the coupling of Cahn-Hilliard system P with the Navier-Stokes equation, such as, for example, in [3]. References [1] F. Boyer and S. injeaud. Numerical schemes for a three component Cahn-Hilliard model. ESAI ath. odel. Numer. Anal., 454: , [2] L. Cherfils,. Petcu, and. Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic oundary conditions. Discrete Contin. Dyn. Syst., 274: , [3] S. Dong. On imposing dynamic contact-angle oundary conditions for wall-ounded liquid-gas flows. Comput. ethods Appl. ech. Engrg., 247/248: , [4] R. Eymard,. Gallouët, and R. Herin. Finite volume methods. Hand. Numer. Anal., VII. North-Holland, Amsterdam, [5] H.P. Fischer, P. aass, and W. Dieterich. Novel surface modes in spinodal decomposition. Phys. Rev. Lett., 79: , Aug [6] H.P. Fischer, P. aass, and W. Dieterich. Diverging time and length scales of spinodal decomposition modes in thin films. EPL Europhysics Letters, 421:49 54, [7] R. enzler, F. Eurich, P. aass, B. Rinn, J. Schropp, E. Bohl, and W. Dieterich. Phase separation in confined geometries: Solving the Cahn Hilliard equation with generic oundary conditions. j-cop-phys-co, 133: , Jan [8] F. Naet. Finite volume analysis for the Cahn-Hilliard equation with dynamic oundary conditions. In Finite volumes for complex applications. VII. ethods and heoretical aspects, Springer Proc. ath. to appear, archives-ouvertes.fr/hal [9] J. Prüss, R. Racke, and S. Zheng. aximal regularity and asymptotic ehavior of solutions for the Cahn-Hilliard equation with dynamic oundary conditions. Ann. at. Pura Appl. 4, 1854: , [10] R. Racke and S. Zheng. he Cahn-Hilliard equation with dynamic oundary conditions. Adv. Differential Equations, 81:83 110, [11] H. Wu and S. Zheng. Convergence to equilirium for the Cahn-Hilliard equation with dynamic oundary conditions. J. Differential Equations, 2042: , 2004.