Unconditionally Energy Stable Numerical Schemes for Hydrodynamics Coupled Fluids Systems

Transcription

1 University of Sout Carolina Scolar Commons Teses and Dissertations 017 Unconditionally Energy Stable Numerical Scemes for Hydrodynamics Coupled Fluids Systems Alexander Yuryevic Brylev University of Sout Carolina Follow tis and additional works at: ttp://scolarcommons.sc.edu/etd Part of te Matematics Commons Recommended Citation Brylev, A. Y.017. Unconditionally Energy Stable Numerical Scemes for Hydrodynamics Coupled Fluids Systems. Doctoral dissertation. Retrieved from ttp://scolarcommons.sc.edu/etd/4010 Tis Open Access Dissertation is brougt to you for free and open access by Scolar Commons. It as been accepted for inclusion in Teses and Dissertations by an autorized administrator of Scolar Commons. For more information, please contact

2 Unconditionally Energy Stable Numerical Scemes for Hydrodynamics Coupled Fluids Systems by Alexander Yuryevic Brylev Bacelor of Arts Hamline University 006 Master of Science New Mexico State University 011 Submitted in Partial Fulfillment of te Requirements for te Degree of Doctor of Pilosopy in Matematics College of Arts and Sciences University of Sout Carolina 017 Accepted by: Xiaofeng Yang, Major Professor Lili Ju, Committee Member Zu Wang, Committee Member Xinfeng Liu, Committee Member Dewei Wang, Outside Committee Member Ceryl L. Addy, Vice Provost and Dean of te Graduate Scool

3 Acknowledgments First of all, I would like to tank my tesis director, professor Xiaofeng Yang, for is support and guidance in writing tis tesis. He eld numerous office ours to work wit me on it, was always prompt in answering s and any questions I ad about te researc. Tis work wouldn t be possible witout im. I also appreciate being funded by a researc grant several times. Next, I tank all my professors during my first two years at te University of Sout Carolina for elping me build te foundations I needed to be able to succeed in my project. Courses in Computational Matematics MATH 708 and MATH 709, taugt by professors Lili Ju and Xiaofeng Yang, as well as Numerical Differential Equations MATH 76 and Applied Matematics MATH 70 and MATH 71, taugt by professor Hong Wang, turned out to be particularly valuable. Finally, I would like to tank te department of Matematics at te University of Sout Carolina for accepting me on a PD program and providing me financial support in te form of teacing assistansip. I really enjoyed my work experience and it was great to be a part of te Gamecock family! ii

4 Abstract Te tesis consists of two parts. In te first part we propose several second order in time, fully discrete, linear and nonlinear numerical scemes to solve te pase-field model of two-pase incompressible flows in te framework of finite element metod. Te scemes are based on te second order Crank-Nicolson metod for time disretizations, projection metod for Navier-Stokes equations, as well as several implicit-explicit treatments for pase-field equations. Te energy stability, solvability, and uniqueness for numerical solutions of proposed scemes are furter proved. Ample numerical experiments are performed to validate te accuracy and efficiency of te proposed scemes tereafter. In te second part we consider te numerical approximations for te model of smectic-a liquid crystal flows. Te model equation, tat is derived from te variational approac of te de Gennes energy, is a igly nonlinear system tat couples te incompressible Navier-Stokes equations and two nonlinear coupled second-order elliptic equations. Based on some subtle explicit-implicit treatments for nonlinear terms, we develop unconditionally energy stable, linear, decoupled time discretization sceme. We also rigorously prove tat te proposed sceme obeys te energy dissipation law. Various numerical simulations are presented to demonstrate te accuracy and te stability tereafter. iii

5 Table of Contents Acknowledgments ii Abstract iii List of Tables v List of Figures vi Capter 1 Numerical analysis of certain scemes for pase field models of two-pase incompressible flows Introduction Te PDE System and Energy Law Second Order, Semi-Discrete Scemes and Teir Energy Stability Fully Discrete Scemes and Energy Stability Numerical Experiments Capter Numerical approximations for smectic A liquid crystal flows Introduction Te smectic-a liquid crystal fluid flow model and its energy law Numerical sceme Numerical Simulations iv

6 Bibliograpy v

7 List of Tables Table 1.1 Table 1. Table 1.3 Caucy convergence test for te linear sceme solving ACNS system; errors are measured in L norm; k grid points in eac direction for k from 4 to 8, = 0., η = 0.1, M = 0.01, λ = 0.001, ν = Caucy convergence test for te linear sceme solving CHNS system; errors are measured in L norm; k grid points in eac direction for k from 4 to 8, = 0., η = 0.1, M = 0.01, λ = 0.001, ν = Caucy convergence test for te nonlinear convex-splitting sceme solving ACNS system; errors are measured in L norm; k grid points in eac direction for k from 4 to 8, = 0., η = 0.1, M = 0.01, λ = 0.001, ν = vi

8 List of Figures Figure 1.1 Temporal evolution of a circular domain driven by mean curvature witout ydrodynamic effects. Te parameters are η = 1.0, M = 1.0, λ = 1.0, = 0.1, Ω = [0, 56] [0, 56] Figure 1. Te areas of te circle as a function of time. η = 1.0, M = 1.0, λ = 1.0, = 0.1, Ω = [0, 56] [0, 56]. Te slope of te line is 6.84 and te teoretical slope is π Figure 1.3 Figure 1.4 Figure 1.5 Snapsots of te relaxation of a square sape by te ACNS system. η = 0.01, λ = M = , = Zero contour plots of te merging and relaxation of two kissing circles by te CHNS system. From left to rigt, t = 0.0, t = 0., t =, t = 4, t = 1, t = 18. η = 0.01, λ = , M = 0.1, ν = 0.1, = Filled contour plots in gray scale of te rising bubble by te CHNS system. From left to rigt, t = 0.64, t = 1., t = 1.6, t =, t =.4, t =.8. η = 0.01, λ = , M = 0.1, ν = 0.01, = 0.01, B = Figure.1 Te L errors of te layer funciton φ, te director field d = d 1, d, te velocity u = u, v and pressure p. Te slopes sow tat te sceme is asymptotically first-order accurate in time. 46 Figure. Snapsots of te layer function φ are taken at t = 0, 0., 0.4 and 0.8 for Example Figure.3 Snapsots of te director field d are taken at t = 0, 0., 0.4 and 0.8 for Example Figure.4 Time evolution of te free energy functional of Example Figure.5 Snapsots of te layer function φ are taken at t = 0, 0.3, 0.4, 0.5, 0.6 and 0.8 for Example vii

9 Figure.6 Snapsots of te director field d are taken at t = 0, 0.3, 0.4, 0.5, 0.6 and 0.8 for Example Figure.7 Snapsots of te profile for te first component uy of te velocity field u = u, v at te center x = and t = 0, 0.45 and viii

10 Capter 1 Numerical analysis of certain scemes for pase field models of two-pase incompressible flows 1.1 Introduction Interfacial problems ave attracted muc attention of scientists for over a century. A classical approac to dealing wit suc problems was to introduce a mes wit grid points on te interfaces wic deforms according to te motion of te boundary. Tis metod, owever, ad a drawback tat large displacement or deformation of internal domains could cause computational issues suc as mes entaglement. To overcome tis, sopisticated remesing scemes were often times used [57]. Oter metods wic proved to work well were te volume-of-fluid VOF [48, 49], te front-tracking [40, 41] and te level-set [61, 78] fixed-grid metods, were te interfacial tension is represented as a body-force or bulk-stress spreading over a narrow region covering te interface. Te VOF metod is a numerical tecnique for tracking and locating te interface between te fluids using te marker function. Te disadvatage of tis metod is in its difficulty maintaining te sarp interface between te fluids and te computation of te surface tension. Te level-set metod as improved te accuracy and, ence, te applicability of te VOF metod. Te problem wit te level-set metod occurs wen one tries to use it in an advection field, for example, uniform or rotational velocity field. In tis case te sape and size of te level set must be conserved, owever, te metod does not guarantee tis, so te level set may get significantly distorted and vanis over several time steps. Tis requires 1

11 te use of ig-order finite difference scemes, suc as ig-order essentially nonoscillatory ENO scemes [47], and even ten, te feasibility of long-time simulations is questionable. To overcome tis difficulty, more sopisticated metods ave been designed, suc as combinations of te level set metod wit tracing marker particles advected by te velocity field [60]. In te front-tracking metod a separate front marks te interface but a fixed grid, only modified near te front to make a grid line follow te interface, is used for te fuid witin eac pase. Pase-field or diffuse-interface model is anoter matematical model for solving various interfacial problems. In recent years it as been successfully used to simulate dynamical processes in many fields and as become one of te major tools to study various systems arising from te energy-based variational formalism. Te metod employs an order parameter, called te pase field, and substitutes boundary conditions at te interface by te partial differential equation involving tis new variable. Te pase field is assigned distinct values on eac pase for example, -1 and 1 and a tin smoot transition layer marking te interface is defined as te set of all points were te pase field takes a certain value for example, 0. Hence te dynamics of te interface can be simulated on a fixed grid witout explicit interface tracking, wic renders te diffuse interface metod an attractive numerical approac to simulate free moving/deforming interfacial problems. Based on variational approaces, te governing system can be derived from te total free energy, wic usually leads to some well-posed nonlinear partial differential equations. Tis makes it possible to carry out matematical analysis and design numerical scemes wic preserve te termo-dynamically consistent dissipation law energy-stable at te discrete level. Te preservation of suc laws is critical for numerical metods to capture te correct long time dynamics. Te dynamics of pase field models can be described by eiter te Allen-Can equation [4] or te Can-Hilliard equation [8, 9] based on coices of Sobolev spaces

12 of te variational approac. In details, te Allen-Can equation is a second-order equation, wic is easier to solve numerically but does not conserve te volume fraction, wile te Can-Hilliard equation is a fourt-order equation wic conserves te volume fraction but is relatively arder to solve numerically. Basically, te coarsegraining macroscopic process described by tese two equations may undergo rapid canges near te interface, so te noncompliance of energy dissipation laws may lead to spurious numerical solutions if te grid and time step sizes are not carefully controlled [5, 36]. Tus, from te numerical point of view, people are particularly interested in designing simple, efficient and energy stable numerical scemes satisfying discrete energy dissipations laws. Tere are several callenges to construct te efficient numerical scemes to solve te ydrodynamics coupled pase field model numerically, namely, i te small interfacial widt introduces tremendous amount of stiffness into te system ii te nonlinear coupling between te pase variable and velocity due to te nonlinear convections and stresses, iii te coupling between te velocity and pressure in te fluid momentum equation. It is by no means an easy task, in particular, te development of any efficient and accurate numerical scemes wile maintaining te dissipative energy law. It is remarkable tat many attempts ave been made in tis direction recently cf. a compreensive summary in [65]. However, due to te complexity of te nonlinear convection terms and stresses in te system, most of developed scemes are eiter only first-order in time [44, 69, 67], or are nonlinear scemes wic need some efficient iterative solvers [76, 15], or only focus on te no flow case [59, 77, 31], or unable to provide te stability analysis [17]. Tere are very few works wit te focus on te development of te second order scemes for te ydrodynamics coupled pase field model. Recently, in [3], a second order, unconditionally stable, semi-discrete sceme for 3

13 te ydrodynamics coupled Can-Hilliard pase field model was developed, wic could be regarded as one of te limited successful efforts in te development of second order scemes. However, in [3], first, te scemes for te computation of te pase field variable are nonlinear tanks to te application of te convex splitting approac, tus one in turn needs some efficient iterative solvers. Second, te computation of te pase variable is always coupled wit tat of te velocity. Tird, te proof of energy law is only for te time discretization case. Terefore, te main objective of tis paper is to develop some fully discrete, second-order, unconditionally stable scemes for te ydrodynamics coupled Can- Hilliard pase field model. We combine several approaces wic ave proved efficient for te pase equations and for te Navier-Stokes equations, namely, linear metods based on te Lagrangian multiplier approac cf. [31] and nonlinear metods based on te convex splitting approac cf. [71, 3, 74,, 35, 75] for te pase equations, and projection-type approaces [5, 43, 30] for te Navier-Stokes equations. For te proposed linear scemes, in spite of te fact tat te computation of te pase field variable is still coupled wit tat of te velocity, one only needs to solve a linear elliptic system. Tis is extremely convenient since one can explicitly find te mass matrix for te linear system associated wit te Finite Element metod or Finite Difference metod. In additions, we prove tat te modified discrete energy law olds for all scemes. For te proposed nonlinear scemes, we prove rigorously its unconditional solvability for te fully discrete case. Ample numerical experiments are performed to validate te accuracy and efficiency tereafter. Te rest of te capter is organized as follows. In Section, we present te wole model and te PDE energy law. In Section 3, we develop te numerical scemes and prove teir unconditional stability and unconditionally unique solvability in te time discrete case. In section 4, te scemes are furter discretized in time and space by mixed finite element approximation. Energy stabilities for fully discrete case are 4

14 proved. For nonlinear scemes, we furter prove te solvability and uniqueness. Finally, we present some numerical experiments to validate our numerical scemes in Section 5. Some concluding remarks are presented in Section Te PDE System and Energy Law. We consider te pase field model for a mixture of two immiscible, incompressible fluids in a confined domain Ω R d, d =, 3. In order to label te two fluids, a pase variable macroscopic labeling function φ is introduced suc tat 1 fluid I, φx, t = 1 fluid II, 1.1 wit a smoot but tin transition layer, wic is controlled by te parameter η 1. Te equilibrium configuration of tis mixing layer, in te neigborood of te level set Γ t = {x : φx, t = 0}, is determined by te microscopic interactions between fluid molecules. For te isotropic interactions, te classical self consistent mean field teory SCMFT in statistical pysics [10] yields te following Ginzburg-Landau type of Helmoltz free energy functional: were te first term contributes to te ydropilic" type tendency of mixing of interactions between te materials and te second part, te double well bulk energy F φ = φ 1 4η represents te ydro-pobic" type tendency of separation of interactions. As a consequence of te competition between te two types of interactions, te equilibrium configuration will include a diffusive interface wit tickness proportional to te parameter η; and, as η approaces zero, we expect to recover te sarp interface separating te two different materials cf., for instance, [80, 6, 1]. Te total energy of te ydrodynamic system is a sum of te kinetic energy E k and te mixing energy E mix : E = E k + E mix = Ω 1 u + λ φ + F φ dx, 1. 5

15 were we assume te density of te two fluids are matced wit ρ = 1 and u is te fluid velocity field. Assuming a generalized Fick s law tat te mass flux be proportional to te gradient of te cemical potential [9, 8, 4, 55], one can derive te following non-conserved Allen-Can-Navier-Stokes ACNS system: φ t + u φ = Mµ, 1.3 µ = δe δφ = λ φ + fφ, 1.4 u t + u u + p ν u = µ φ, 1.5 u = 0, 1.6 were fφ = F φ = φ 1φ, p is te pressure, M is te relaxation or mobility η parameter of te pase function, and ν is te viscosity parameter. If te variational derivative can be taken in te H 1, leading to te conserved Can-Hilliard-Navier- Stokes CHNS equations, φ t + u φ = M µ, 1.7 µ = λ φ + fφ. 1.8 Trougout te paper, we assume te boundary conditions u Ω = 0, n φ Ω = 0, n µ Ω = 0, 1.9 altoug all results are valid for periodic boundary conditions as well. Since te above system was derived from te energetic variational formulation, it can be readily establised tat te total energy of te ACNS system , and CHNS system are dissipative. More precisely, by taking te inner product of 1.3 wit E, 1.5 wit u, and ten summing up tese equalities, φ we obtain te following energy dissipation law for ACNS system: d dt E = Ω ν u E + M dx φ 6

16 For CHNS system, by taking te L inner product of 1.7 wit µ, of 1.8 wit φ t, of 1.5 wit u and summarize all equalities, we obtain d dt Ω E = ν u + M µ dx However, for Allen-Can equation 1.3, te variational derivative δe δφ involves te second order derivative, tus it is not suitable to use tem as test functions in numerical approximations, making it difficult to prove te discrete energy dissipation law. Hence, it is a common practice to rewrite as te following equivalent form [79, 37, 65]. u t + u u ν u + p + 1 M φ t + u φ φ = 0, 1.1 u = Now, by taking te L inner product of 1.3 wit φ t and 1.1 wit u, we obtain d dt Ω E = ν u + 1 M φ dx, wit φ = φt + u φ We empasize tat te above derivation is suitable in a finite dimensional approximation since test function φ t is in te same subspaces as φ. Hence, it allows us to design numerical scemes wic satisfy a discrete energy law. Remark It is well known tat te solutions of te conserved Can Hilliard pase equation, wit suitable boundary conditions, satisfy te desired conservation property t Ω φdx = 0, wic is not satisfied by te solutions of te non-conserved Allen Can equation. In fact, one can add a scalar Lagrange multiplier in 1.3 to enforce tis conservation property cf. [79, 68], or modify te free energy functional by adding a penalty term for volume, similar as [18]. Bot ways will not introduce any matematical or numerical difficulty, tus we sall not include it in te discussions below. 7

17 For simplicity, we consider only in tis paper te ACNS model All teoretical proof can be generalized to te CHNS model witout any furter difficulty. Te detailed stability proof for CHNS system will be left to te interested readers. 1.3 Second Order, Semi-Discrete Scemes and Teir Energy Stability. In tis section, we construct several second order in time, semi-discrete scemes and present teir energy stabilities. Let > 0 be a time step size and set t n = n for 0 n N = [T/]. Witout ambiguity, we denote by fx, gx = Ω fxgxdx 1 te L inner product between functions fx and gx, by f = f, f te L norm of function fx Te Linear Sceme. We first construct a linear sceme based on a Lagrange multiplier approac in [31], were it is first developed to solve te Can-Hilliard equation witout flow. A function q = φ 1 η is introduced suc tat one can write fφ = qφ. It ten follows tat q t = η φφ t. By using te variable q, te total energy can be written as E = Ω 1 u + λ φ + η 4 q dx It is remarkable tat energy dissipation laws 1.11 and 1.14 still old. A linear sceme for solving te ACNS system is constructed as follows: Given te initial conditions φ 0, u 0, q 0 = φ0 1 η and p 0 = 0, we compute φ 1, u 1, q 1 and p 1 by any first order metods cf. [31, 69, 70]. Having computed φ n 1, q n 1, u n 1, p n 1 and φ n, q n, u n, p n for n 1, we compute φ n+1, q n+1, ũ n+1, u n+1, p n+1 by te following steps. 8

18 Step 1: φ n+1 λm φn+1 + φ n = 3 3 φn 1 q n+1 + q n φn 1 = 0, 1.16 η q n+1 q n φn 1 φ n+1 φ n φn 1, 1.17 ũ n+1 u n ν ũ n+1 + u n 3 + B un 1 un 1, ũ n+1 + u n + p n n+1 φ + M 3 φn 1 φn 1 = 0, 1.18 wit boundary conditions n φ n+1 Ω = 0, ũ n+1 Ω = 0, n q n+1 Ω = 0, 1.19 were Step : Bu, v := u v + 1 uv, 1.0 φ n+1 = φn+1 φ n + ũ n+1 + u n 3 φn 1 φn u n+1 ũ n pn+1 p n = 0, 1. u n+1 = 0, 1.3 u n+1 n Ω = Several remarks are in order. Remark In fact, 1.17 can be rewritten as q n+1 = q n + η φ n+ 1 φ n+1 φ n, 1.5 were φ n+ 1 = 3 φn 1 φn 1. Tus we can replace te q n+1 in 1.16, it is equivalent to te following. φ n+1 λm φn+1 + φ n + λm φ n n+ φ η φ n+1 φ n + q n =

19 Terefore, we can solve for φ n+1 and ũ n+1 directly from 1.6 and Once we obtain φ n+1, te q n+1 is automatically given in 1.5. Namely, te new variable q does not involve any extra computational costs. Remark We note tat q n+1 is formally a second order approximation of φ 1 η. Indeed, Eq implies tat q n+1 φn φ n+1 φ n = q n φn φ n φ n 1 + R η η η η n+1 1.7, were R n+1 = 1 η φ n+1 φ n + φ n 1. Since, euristically, R k = O 4 for 0 k n and assuming q 1 φ φ 1 φ 0 η η order approximation, we ten get q n+1 φn φ n+1 φ n η η = O for instance, by a first = O. Notice tat φ n+1 φ n O, terefore, q n+1 is formally a second order approximation to φ 1 η. Remark A second order pressure correction sceme [43] is used to decouple te computations of pressure from tat of te velocity. Tis projection metods are analyzed in [66] were it is sown discrete time, continuous space tat te scemes are second order accurate for velocity in l 0, T ; L Ω but only first order accurate for pressure in l 0, T ; L Ω. Te loss of accuracy for pressure is due to te artificial boundary condition 1. imposed on pressure [0]. We also remark tat te Crank-Nicolson sceme wit linear extrapolation is a popular time discretization for te Navier-Stokes equation. We refer to [39] and references terein for analysis on tis type of discretization. Remark Bu, v is te skew-symmetric form of te nonlinear advection term in te Navier-Stokes equation, wic is first introduced by Temam [7]. If te velocity is divergence free, ten Bu, u = u u. We define bu, v, w := Bu, v, w

20 In our numerical sceme ũ n+1 +u n is not divergence free, but notice te following identity bu, v, v = Bu, v, v = 0, if u n Ω = In oter words, tis identity olds regardless of weter u or v are divergence free or not, wic would elp to preserve te discrete energy stability. Remark It is remarkable tat in [70], te autors proposed some linearized scemes for Allen-Can equation no flow case using te second order backward differentiation formulas BDF, were te nonlinear term fφ is treated by second order extrapolation. However, te linear second order sceme in [70] is conditionally stable were tere exists a constraint on te time step. Te autors in [70] also developed a second order, unconditional stable sceme based on te Crank-Nicolson metod, owever, te obtained scemes are nonlinear. In [77], te autors developed a second order, linear, unconditionally stabilized sceme for Can-Hilliard equation no flow case based on te convex splitting approac for te modified functional F φ te functional F φ is modified to get uniform upper bound for its second order derivative. However, a ig order stabilizer term φ n+1 φ n for Can-Hilliard equation, analogously φ n+1 φ n for Allen-Can equation is added in teir sceme tat as iger splitting error O φ t H from spatial derivatives, tat is somewat not resonable because te splitting error is muc iger tan te explicit treatment for te nonlinear term of fφ n. Te framework of te above sceme takes te second order Crank-Nicolson to discretize te pase equation Inspired from [31], we introduce a new variable qx tus te order of Ginzburg-Landau double well potential is reduced by alf. By some explicit-implicit treatments, we obtain a linear sceme wile maintaining te second order accuracy As we sall 11

21 sow below, te above sceme is unconditionally energy stable. It is noticable te discrete energy we obtained is te modified energy 1.31, were te nonlinear potential F φ is replaced by te term of q. We empasize tat te obtained dissipation law in 1.30 is a second order approximation of te PDE energy law 1.14 note tat te energy law is =" in stead of ". To te best of te our knowledge, tis is te first suc unconditionally stable, second order, and linearized sceme for te ydrodynamics coupled pase field model. Teorem Te solution of satisfies te following discrete energy law Eu n+1, φ n+1, q n pn+1 were = Eu n, φ n, q n + 8 pn 1 M φ n+1 ũ + ν n+1 + u n 1.30, Eu, φ, q = 1 u + λ 1 φ + η 4 q, 1.31 tus te sceme is unconditionally stable. Proof. By taking te L inner product of 1.16 wit φn+1 φ n, and performing integration by parts, we obtain M 1 M φ n+1 1 φ n+1, ũ n+1 + u n 3 M φn 1 φn 1 1 φn+1 1 φn + λ + λ 3 = 0. φn 1 q n+1 + q n φn 1, φ n+1 φ n 1.3 By taking te L inner product of 1.17 wit λ qn+1 +q n, we obtain λη 4 qn+1 q n λ 3 φn 1 q n+1 + q n φn 1, φ n+1 φ n =

22 By taking te L inner product of 1.18 wit ũn+1 +u n, and using identity 1.9, we obtain 1 ũn+1 u n ũ + ν n+1 + u n + p n, ũn+1 + u n + 1 φn+1 3 M φn 1 φn 1, ũn+1 + u n = 0, 1.34 By taking te L inner product of 1. wit u n+1 and performing integration by parts, we ave 1 un+1 ũ n+1 + u n+1 ũ n+1 = 0, 1.35 were we use explicitly te divergence-free condition for u n+1, p n+1 p n, u n+1 = p n+1 p n, u n+1 = 0. We rewrite te projection step 1. as 1 u n+1 + u n ũ n+1 + u n + 1 pn+1 p n = 0. By taking te inner product of te above equation wit pn, one arrives at p n+1 p n p n+1 p n = p n, ũ n+1 + u n On te oter and, it follows directly from 1. tat 8 pn+1 p n = 1 un+1 ũ n Hence, by combining , we obtain 1 M φ n+1 + λ 1 φn+1 1 φn + λ η 4 qn+1 q n + 1 un+1 u n ũ + ν n+1 + u n + p n+1 p n 8 = Tis concludes te proof. 13

23 Remark It is obvious tat 1 Eu n+1, φ n+1, q n pn+1 Eu n, φ n, q n 8 pn is a second order approximation of d Eu, φ, q at tn+ 1. Te similar sceme can be applied to te CHNS model, te sceme reads as follows. Step 1: Step : φ n+1 φ n + ũ n+1 + u n 3 φn 1 φn 1 M µ n+ 1 = 0, 1.39 µ n+ 1 = λ φn+1 + φ n 3 φn 1 q n+1 + q n φn 1, 1.40 η q n+1 q n = 3 φn 1 φ n+1 φ n φn 1, 1.41 ũ n+1 u n ν ũ n+1 + u n 3 + B un 1 un 1, ũ n+1 + u n + p n + µ n+1/ 3 φn 1 φn 1 = 0, 1.4 n φ n+1 Ω = 0, n µ n+1/ Ω = 0, ũ n+1 Ω = u n+1 ũ n pn+1 p n = 0, 1.44 u n+1 = 0, 1.45 u n+1 n Ω = Similary, we obtain te energy stability as follows. Te detailed proof is left to te interested readers. Teorem Te solution of satisfies te following discrete energy law Eu n+1, φ n+1, q n pn+1 = Eu n, φ n, q n + 8 pn M µ n+ 1 ũ ν n+1 + u n. were Eu, φ, q = 1 u + λ 1 φ + η 4 q, tus te sceme is unconditionally stable. 14

24 1.3. Te Nonlinear Sceme. We now propose a second order, semi-discrete numerical sceme to solve te ACNS system based on te convex-splitting approac for te nonlinear potential F φ. A similar sceme for solving te CHNS system ad been proposed in [3]. For te nonlinear potential, we can rewrite F φ as te sum of a convex function and a concave function as F φ = F v φ + F c φ := 1 4η φ η φ + 1, and accordingly fφ = F vφ + F cφ. Te idea of convex-splitting is to use explicit discretization for te concave part i.e. F c 3 φn 1 φn 1 and semi-implicit discretization for te convex part. Furter we approximate F v φn+1 +φ n by te Crank-Nicolson sceme F v φn+1 + φ n F vφ n+1 F v φ n = 1 φ n+1 + φ n φ n+1 + φ n. φ n+1 φ n η Suc a second order convex-splitting sceme is originally proposed and analyzed in [35, ] in te context of pase field crystal equation, see also [71] for applications in tin film epitaxy. Having computed φ n 1, q n 1, u n 1, p n 1 and φ n, q n, u n, p n for n 1, we compute φ n+1, q n+1, ũ n+1, u n+1, p n+1 by te following steps. were Step 1: φ n+1 λm φn+1 + φ n ũ n+1 u n ν ũ n+1 + u n + p n + f 0 φ n+1, φ n = 0, B un 1 un 1, ũ n+1 + u n n+1 φ M 3 φn 1 φn 1 = 0, 1.48 n φ n+1 Ω = 0, ũ n+1 Ω = φ n+1 = φn+1 φ n + ũ n+1 + u n 3 φn 1 φn 1,

25 Step : f 0 φ n+1, φ n = 1 φ n+1 + φ n φ n+1 + φ n η 1 η 3 φn 1 φn u n+1 ũ n pn+1 p n = 0, 1.5 u n+1 = 0, 1.53 u n+1 n Ω = We sow te energy stability teorem as follows. Teorem Te solution of te sceme satisfies te discrete energy law of Eu n+1, φ n+1 + λ 4η φn+1 φ n + 8 pn+1 Eu n, φ n + λ 4η φn φ n pn 1 M φ n+1 ũ + ν n+1 + u n, were Eu, φ = 1 u 1 + λ φ + F φ, 1, tus te sceme is unconditionally stable. Proof. Te only difference between te linear sceme and convex splitting sceme is in te discretization of te Allen-Can equation. By taking te L inner product of 1.47 wit φn+1 φ n, and performing integration M by parts, one obtains 1 M φ n+1 1 φ n+1, ũ n+1 + u n 3 M φn 1 φn 1 + λ 1 φn+1 1 φn + λ 1.55 f 0 φ n+1, φ n, φ n+1 φ n = 0. Recall te definition of f 0 φ n+1, φ n from 1.51 and te equality as follows, 1 3φ n φ n 1, φ n+1 φ n = 1 φ n+1 + φ n, φ n+1 φ n 1 φ n+1 φ n + φ n 1, φ n+1 φ n = 1 φ n+1 φ n 1 4 φ n+1 φ n φ n φ n 1 + φ n+1 φ n + φ n 1 16

26 we obtain 1 M φ n+1 1 φ n+1, ũ n+1 + u n 3 M φn 1 φn 1 + λ 1 φn+1 1 φn + F φ n+1 F φ n, λ 1 φ n+1 φ n φ n φ n 1 + φ n+1 φ n + φ n 1 = 0. 4η For te momentum equation, we get te same results as Tus, by combining wit 1.56, we obtain 1 M φ n+1 + λ 1 φn+1 1 φn + F φ n+1 F φ n, un+1 u n ũ + ν n+1 + u n + p n+1 p n 8 + λ 1 φ n+1 φ n φ n φ n 1 + φ n+1 φ n + φ n 1 4η 1.57 = 0. Tis completes te proof. Remark Heuristically, Eu n+1, φ n+1 + λ 4η φ n+1 φ n + 8 pn+1 is a second order approximation of Eu n+1, φ n+1, as one can write φ n+1 φ n = φ n+1 φ n /, and φ n+1 φ n / is an approximation of φ t at t n+1/. 1.4 Fully Discrete Scemes and Energy Stability. We now consider te fully discrete versions of scemes and to solve te system in te framework of finite element metod. Let T be a quasi-uniform triangulation of te domain Ω of mes size. We introduce X and Y te finite element approximations of H0Ω 1 and H 1 Ω respectively based on te triangulation T. In addition, we define M = Y L 0Ω := {q Y ; Ω q dx = 0}. We assume tat X and M are stable approximation spaces for 17

27 te velocity and pressure in te sense tat tere exists a constant c suc tat v, q sup v X v H 1 c q L, q M. It is pointed out [8] tat te inf-sup condition is necessary for te stability of pressure even toug one may solve te projection step as a pressure Poisson equation. For simplicity, te following notations will be used tereafter. φ n+ 1 ũ n+ 1 = φn+1 = ũn+1 + φ n + u n f 0 φ n+1, φ n = 1 η 1, φ n+ 1, u n+ 1 = 3φn φ n 1, 1.58a = 3un u n 1 φ n+1 + φ n φ n+ 1 φ n+ 1, 1.58b. 1.58c Te fully discrete Linear sceme. We now give te fully discrete formulation for te linear sceme In te framework of te finite element spaces above, te sceme reads as follows. Find φ n+1, q n+1, ũ n+ 1, p n+1, u n+1 Y Y X M X suc tat for all ψ, v, g Y X Y tere old Step 1: φ n+1 φ n, ψ + ũ n+ 1 φ n+ 1 n+, ψ + λm φ 1, ψ + λm φ n η φn+ φ n+1 φ n + q, n ψ 1.59 = 0, ũ n+ 1 u n n+, v + ν ũ 1 n+, v + b u 1, ũ n+ 1, v = p n 1, v φ n+1 φ n + ũ n+ 1 φ n+ 1 φ n+ 1, v M Step : u n+1 ũ n+1 1, v + p n+1 p n, v = 0, 1.61 u n+1, g =

28 Step 3: η q n+1 q n, ψ = φ n+ 1 φ n+1 φ n, ψ Remark Note tat te update of q n+1 equations. in 1.63 is decoupled from te rest of Remark We remark tat te velocity u n+1 is sougt in te space X H 1 0Ω. It implies tat u n+1 satisfies te essential boundary condition u n+1 = 0 on Ω, wereas u n+1 H satisfies u n+1 n = 0 on Ω. Here H := {v L Ω, v = 0, u n+1 n Ω = 0}. As is pointed out in [9] Remark 3.3, X is a discrete approximation of H since H 1 0 is dense in H. Tus 1.61 and 1.6 can be viewed as an approximation of te Darcy problem. Tis formulation is sown in [9] to yield an optimal condition number for te pressure operator associated wit te finite element spatial discretization. In order to establis te stability of te fully discrete sceme , for convenience, we introduce te discrete negative divergence operator B : X H 1 0 M H 1, endowed wit L norm suc tat for u X and q M B u, q := u, q = u, q := u, B T q, 1.64 were B T : M X is te transpose of B te discrete gradient operator. Tus one can write te projection step 1.61 and 1.6 in te discrete form u n+1 ũ n+1 + BT p n+1 p n = 0, in X, 1.65 B u n+1 = 0, in M Now one can proceed by testing te above equation wit u n+1, B T p n+1 X, respectively. Te modified energy law is valid wit te discrete gradient operator associated wit pressure terms. 19

29 Teorem Given tat q n Y, φ n, φ n 1 Y, u n, u n 1 X, and p n M, te system admits a unique solution φ n+1, q n+1, ũ n+ 1, p n+1, u n+1 Y Y X M X at te time step t n+1 for any > 0 and > 0. Moreover, te solution satisfies a discrete energy law Eu n+1, φ n+1, q n BT p n+1 Eu n, φ n, q n + φ n ν ũ n+ 1 M φ n+1 8 BT p n + ũ n+ 1 φ n+ 1, 1.67 were Eu n, φ n, q n = 1 un + λ 1 φn + η 4 qn. Tus te sceme is unconditionally stable. Proof. We note tat te sceme is a linear system. Tus te unique solvability would follow from te energy law To establis te energy law, we define an intermediate variable q n+1 suc tat q n+1 = η φn+ 1 φ n+1 φ n + q n. Ten 1.59 and 1.63 can be written as φ n+1 φ n η q n+1 q n, ψ + ũ n+ 1 φ n+ 1, ψ +λm φ n+ 1, ψ + λm φ n+ 1, ψ = φ n+ 1 φ n+1 q n+1 + q n, ψ = 0, 1.68 φ n, ψ, 1.69 q n+1, ψ = q n+1, ψ One can obtain te following version of energy law, by working wit 1.68, 1.60, 1.61, 1.6, 1.65, 1.66, and 1.69, and by following te same proof of Teorem 1.3.1, Eu n+1, φ n+1, q n+1 = ν ũ n BT p n+1 Eu n, φ n, q n + 8 BT p n φ n φn+1 M + ũ n+ 1 φ n

30 Here Eu n, φ n, q n = 1 un + λ 1 φn + η 4 qn. Te energy law 1.67 ten follows from te fact q n+1 Ł q n+1 Ł as is evident from Te proof of te teorem is complete. For completeness, we also gives te corresponding linear sceme for solving te CHNS system: Find φ n+1, q n+1, µ n+ 1, ũ n+ 1, p n+1, u n+1 Y Y Y X M X suc tat for all ψ, ϕ, v, g Y Y X Y tere olds Step 1: φ n+1 φ n, ψ + M µ n+ 1, ψ φ n+ 1 ũ n+ 1 n+ µ 1 n+1/, ϕ = λ φ, ϕ + λ φ n+ 1, ψ = 0, η φn+ φ n+1 φ n + q, n ϕ, 1.73 ũ n+ 1 u n, v + ν ũ n+ 1, v + b u n+ 1, ũ n+ 1, v = p n, v λ φ n+ 1 µ n+ 1, v Step : u n+1 ũ n+1, v + p n+1 p n, v = 0, 1.75 u n+1, g = Step 3: η q n+1 q n, ψ = φ n+ 1 For te sceme , one can prove Teorem Given tat q n, φ n, φ n 1 φ n+1 φ n, ψ Y, u n, u n 1 X, and p n M, te system is uniquely solvable, for any > 0 and > 0. Moreover, te solution satisfies a discrete energy law Eu n+1, φ n+1, q n BT p n+1 Eu n, φ n, q n + 8 BT p n 1

31 M µ n+ 1 ν ũ n+ 1, were Eu n, φ n, q n = 1 un + λ 1 φn + η 4 qn. Tus te sceme is unconditionally stable. Remark We point out tat te modified energy stability in Teorem implies te discrete L stability for u n+1, φ n+1 and q n+1 in te sceme for te ACNS model. Te H 1 stability for φ n+1 is not yet available, altoug suc an estimate is valid for te original PDE as is implied by te continuous energy law Eq We note tat formally q n+1 of φ 1 η, as is explained in Remark is a second order in-time approximation Tis is in contrast to te case of CHNS model. We note tat te sceme is mass-conservative, in te sense tat Ω φn+1 dx = Ω φn dx = = Ω φ0 dx. Hence te L stability of φ n+1 in te sceme plus Poincare inequality implies te H 1 stability of φ n+1. Tese remarks are also true for te semi-discrete linear scemes ACNS and CHNS. Note tat te H 1 stability of φ n+1 are valid in te nonlinear scemes for bot ACNS and CHNS models, see Teorem and Proposition below. Remark Te error analysis of te discrete scemes and can be very difficult. For instance, te error analysis of Eq would require an L estimate of te derivative φn+1 φ n+1 φ n, wic in turn needs ig order estimate of via Eq among oters. Te case for CHNS could be even worse because of a lack of diffusion in q n+1 for te ig order estimates of φ n+1. We leave te error analysis of tese scemes to a future work Te Fully Discrete Nonlinear Sceme. Now we present te fully discrete version of te nonlinear sceme of Te sceme reads as follows.

32 Find φ n+1, ũ n+ 1, p n+1, u n+1 Y X M X suc tat for all ψ, v, g Y X Y, tere old, Step 1: φ n+1 φ n, ψ + ũ n+ 1 φ n+ 1 + λm 4η n+1/, ψ + λm φ, ψ φ n+1 + φ n φ n+1 + φ n, ψ = λm η φ n+ 1, ψ, 1.78 ũ n+ 1 u n n+, v + ν ũ 1 n+, v + b u 1, ũ n+ 1, v = p n 1, v φ n+1 φ n + ũ n+ 1 φ n+ 1 n+ φ 1, v M Step : u n+1 ũ n+1 1, v + p n+1 p n, v = 0, 1.80 u n+1, g = Te following teorem states tat te fully discrete sceme is unconditionally uniquely solvable and satisfies a discrete energy law. Teorem Given tat φ n, φ n 1 Y, u n, u n 1 X, and p n M, te system admits a unique solution φ n+1, ũ n+ 1, p n+1, u n+1 Y X M X at te time step t n+1 for any parameters > 0 and > 0. Moreover, te solution satisfies a discrete energy law Eu n+1, φ n+1 + λ 4η φn+1 Eu n, φ n + λ ν ũ n+ 1 φ n + 8 BT p n+1 were Eu n, φ n = 1 un 1 + λ φn + F φ n, 1 4η φn φ n BT p n φ n+1 φ n + ũ n+ 1 φ n+ 1, M. Tus te sceme is unconditionally stable. 3

33 Proof. Note tat 1.80 and 1.81 or equivalent 1.65 and 1.66 are decoupled from te rest of te system. Given ũ n+1 or equivalent ũ n+ 1, te unique solvability of 1.80 and 1.81 is classical, see for instance [9]. Hence one only needs to sow tat 1.78 and 1.79 are uniquely solvable. We define a finite dimensional Hilbert space Z := Y X endowed wit te usual H 1 inner product and norm. We introduce an operator S : Z Z suc tat S φ, u, ψ, v = 1 φ φ n Z M + λ + u φ n+ 1, ψ + λ φ + φ n φ + φ n, ψ 4η u u n φ + φ n, ψ λ n+ φ 1 η, ψ +, v + ν n+ u, v + b u 1, u, v + p n 1, v + φ φ n + u φ n+ 1 n+ φ 1, v, M for φ, u Z and ψ, v Z. Note tat Eq is scaled by 1 in te defini- M tion of S. It is clear from Sobolev embedding and Hölder s inequality tat te operator S is a continuous operator. We proceed to sow tat S φ, u, φ, u Z > 0, for φ, u Z large enoug. Ten Lemma 1.4 in [73] pp. 164 implies tat tere exists φ n+1, ũ n+ 1 Z suc tat S φ n+1, ũ n+ 1 = 0. We ave 1 φ φ n + u φ n+ 1, φ + 1 φ φ n + u φ n+ 1 n+ φ 1, u M M = 1 φ φ n + u φ n+ 1 1 φ φ n + + u φ n+ 1, φn M M 1 φ φ n + u φ n+ 1 1 φ + φ n + φ M M φn 1 M u L 4 φ n+ 1 φn L 4 1 φ φ n + u φ n+ 1 1 φ φ + n φ + M M φn ν u Cν, λ, M, φ n+ 1 φ n H 1, were one as utilized te Sobolev embedding and te Poincaré inequality. Furter- 4

34 more, applying te Young s inequality and Sobolev embedding, we can obtain λ φ + φ n 4η = λ 4η λ 8η Ω Ω φ + φ n, φ φ 4 φ n 4 dx + λ 4η φ 4 dx Cη, λ, φ n 4 H 1. φ + φ n φ + φ n, φ n Combing te above inequalities, and in view of te skew symmetry 1.9 of te trilinear form bu, v, w, we obtain S φ, u, φ, u C φ H + u 1 H 1 Z Cν, λ, M, p n + φ n 4 H + 1 un + φ n Tus for φ, u Z large enoug, one as S φ, u, φ, u > 0. Te existence of φ n+1, ũ n+ 1 Z Y X to 1.78 and 1.79 are ence proved. For uniqueness, suppose φ n+1,i, 1 ũn+,i, i = 1, are two solutions of 1.78 and Ten teir differences φ = φ n+1,1 φn+1, and ũ = ũ n+ 1,1 ũn+ 1, satisfy φ, ψ + + Mλ 8η = 0, ũ φ n+ 1 Mλ, ψ + φ, ψ φ φ n+1,1 + φn+1, + φ n + φ n+1,1 + φ n+1, + φn, v and ũ, v + ν ũ, v + b u n+ 1, ũ, v + 1 φ M + ũ φ n+ 1 φ n , v 1.86 = Te uniqueness follows simply by taking te test functions ψ = 1 M φ and v = ũ in 1.84 and 1.87 respectively, and summing up te results. 5

35 Finally, for te stability of te sceme, one works wit 1.78, 1.79 and abstract equations 1.65, Following te same procedure as in te proof of Teorem 1.3.3, one derives te modified energy law 1.8. Tis concludes te proof. We comment ow to implement te nonlinear sceme Note tat te only nonlinear term appears in te Allen-Can equation We tus adopt a Picard iteration procedure on velocity to decouple te computation of te nonlinear Allen-Can equation 1.78 from tat of te linear Navier-Stokes equation Denote by i te Picard iteration index. Specifically, given te velocity ũ n+ 1,i, we solve for φ n+1,i+1 from te Allen-Can equation 1.78 by Newton s metod. As φ n+1,i+1 is available, we can ten proceed to solve for ũ n+ 1,i+1 from te linear equations We repeat tis procedure until te relative difference between two iterations witin a fixed tolerance. Numerical simulations in [46] suggest tat at least 4 grid elements across te interfacial region of tickness η are needed for accuracy. To improve te efficiency of te algoritm, we explore te capability of adaptive mes refinement of FreeFem++ cf. [33] in wic a variable metric/delaunay automatic mesing algoritm is implemented. Finally we state witout a proof tat similar result as Teorem also olds for te fully discrete sceme for solving CHNS system see in [3]. Proposition Given tat φ n, φ n 1 Y, u n, u n 1 X, and p n M, tere exists a unique solution φ n+1, µ n+ 1, ũ n+ 1, p n+1, u n+1 Y Y X M X to te corresponding system for solving te CHNS system for any parameters > 0 and > 0. Moreover, te solution satisfies a discrete energy law Eu n+1, φ n+1 + λ 4η φn+1 φ n + 8 BT p n+1 Eu n, φ n + λ 4η φn φ n BT p n 6

36 Tus te sceme is unconditionally stable. M µ n+ 1 ν ũ n Numerical Experiments In tis section, we present some numerical results using our scemes. We use P1 P1 or P P finite element function spaces for Y Y, and P1b P1 or Taylor- Hood P P1 mixed finite element spaces for X Y. It is well-known tat tese approximation spaces satisfy te inf-sup conditions for te biarmonic operator and Stokes operator, respectively cf. [14, 1] Convergence Tests. In tis subsection, we verify te second order convergence of te proposed scemes via te Caucy convergence test. Te computational domain is [0, 1] [0, 1], we take uniformly k grid points in eac direction for k from 4 to 8 = k, and we take a linear refinement pat = 0.. We calculate te rate at wic te Caucy difference e.g. φ k φ k 1 converges to zero in te L norm at te final time T = 0.1. Here te P1 finite element space is used for Y, and te mini P1b P1 mixed finite element spaces are used for X Y. Te error is expected to be at te order of e = o + o = o. For all of te convergence tests, te initial conditions are taken to be φ 0 = 0.4 cosπx cosπy cosπx cos3πy, u 0 = sinπx sinπy, sinπy sinπx. Te parameters of te problem are η = 0.1, M = 0.01, λ = 0.001, ν = 0.1. Te errors and convergence rates are presented in Tables 1.1, 1., and 1.3 for te linear sceme solving ACNS, te linear sceme solving CHNS and nonlinear sceme solving ACNS, respectively. Te results sow 7

37 tat te scemes are of second order accuracy for φ and u in L norm, and te rate of convergence for pressure p appear to be only first order wic is known for te pressure projection sceme. Table 1.1: Caucy convergence test for te linear sceme solving ACNS system; errors are measured in L norm; k grid points in eac direction for k from 4 to 8, = 0., η = 0.1, M = 0.01, λ = 0.001, ν = rate 3 64 rate rate φ 1.50e e e 5 u 3.00e e e e 5 v 3.00e e e e 5 p 1.65e e e e 4 Table 1.: Caucy convergence test for te linear sceme solving CHNS system; errors are measured in L norm; k grid points in eac direction for k from 4 to 8, = 0., η = 0.1, M = 0.01, λ = 0.001, ν = rate 3 64 rate rate φ 3.31e e e 4 u 3.00e e e e 5 v 3.10e e e e 5 p 1.65e e e e 4 Table 1.3: Caucy convergence test for te nonlinear convex-splitting sceme solving ACNS system; errors are measured in L norm; k grid points in eac direction for k from 4 to 8, = 0., η = 0.1, M = 0.01, λ = 0.001, ν = rate 3 64 rate rate φ 1.60e e e 4 u 3.00e e e e 5 v 3.00e e e e 5 p 1.65e e e e Te Velocity of a Circular Moving Interface. Here we perform a classical numerical experiment of a srinking circular bubble [13] for Allen-Can equation u = 0, no flow. We sow tat we can accurately 8

38 t = 1000 t = 000 t = 3000 t = 4000 t = 5000 Figure 1.1: Temporal evolution of a circular domain driven by mean curvature witout ydrodynamic effects. Te parameters are η = 1.0, M = 1.0, λ = 1.0, = 0.1, Ω = [0, 56] [0, 56]. 3.5 x area time Figure 1.: Te areas of te circle as a function of time. η = 1.0, M = 1.0, λ = 1.0, = 0.1, Ω = [0, 56] [0, 56]. Te slope of te line is 6.84 and te teoretical slope is π. calculate te velocity of a moving interface via our second order scemes. Te set-up of te numerical experiment is te same as of tat in [13] wit te domain size of Ω = [0, 56] [0, 56]. Te parameters are taken to be unity, i.e., η = 1, M = 1, λ = 1. Initially, tere is a circular interface boundary of radius 100 in te middle of Ω. Witin te circle te pase variable is +1 and outside it takes te value 1. As it is pointed out in [13], te circular boundary will srink and disappear eventually driven by te mean curvature. Te area of te circle can be found explicitly, A = A 0 πt, were A 0 is te initial area. Tus te circular boundary srinks at a rate of V = π. Te temporal evolution of te circular domain is sown in Fig We take 9

39 = 0.1 in te simulation. In space, we explore te adaptive mes refinement of FreeFem++ cf. [33] suc tat at least four grid cells are located across te diffuse interface min 0.. Te results sown are produced using te linear sceme , and te same results are obtained via te nonlinear convex splitting sceme. Te areas of te circle as a function of time is sown in Fig. 1.. Te slope of te line is approximately Te relative error of te velocity compared to te teory is 1.61e 04. Our sceme performs better tan any of te scemes presented in [13] in tis regard Sape relaxation. In tis subsection, we perform anoter two standard numerical tests in te context of pase field fluid models sape relaxation in Fig. 1.3 and Fig In bot tests, te P finite element space is used for Y, and te Taylor-Hood P P1 mixed finite element spaces are used for X Y. t = 4 t = 40 t = 60 t = 80 Figure 1.3: Snapsots of te relaxation of a square sape by te ACNS system. η = 0.01, λ = M = , = In te first numerical test, we simulate te square evolves to a circular bubble using te linear sceme for te ACNS system. Te pase variable φ takes value +1 inside te square and 1 outside of it. Te initial velocity u and pressure p are zeros. Te parameters are η = 0.01, λ = M = , ν = 0.1, = Fig. 1.3 sows some snapsots of te zero contour of te order parameter 30

40 of te ACNS system. We observe tat te isolated square sape relaxes to a circular sape, due to te effect of surface tension. Te sape also appears to srink a bit during te relaxation because te ACNS system is not mass-conservative. In te second numerical test, we simulate te merging process of two circular bubbles using te CHNS system. In Fig. 1.4, we sow te merging of two circles next to eac oter under te influence of surface tension in te CHNS system. Te circles quickly connect and eventually relax to a large circle at wic te surface energy is minimal. In tis simulation, te parameters are η = 0.01, λ = , M = 0.1, ν = 0.1, = Figure 1.4: Zero contour plots of te merging and relaxation of two kissing circles by te CHNS system. From left to rigt, t = 0.0, t = 0., t =, t = 4, t = 1, t = 18. η = 0.01, λ = , M = 0.1, ν = 0.1, = A Rising Bubble. In tis numerical experiment, we simulate te rising process of a ligter liquid bubble using te CHNS system driven by te buoyancy. In particular, te density difference of te two fluids is small so tat a Boussinesq approximation is applicable [56]. Specifically, a buoyancy term Gρφ ρŷ := Bφ φŷ is added to Navier- 31

41 Stokes equation. Here ρφ = 1+φ ρ φ ρ, ρ and φ are te spatial averages of ρ and φ, B = G ρ 1 ρ, ŷ = 0, 1 T. Te filled contour plots in gray scale of te rising bubble are sown in Fig We see tat te bubble rises due to te buoyancy. As it rises, te bubble also elongates orizontally, especially wen it is near te upper boundary. Figure 1.5: Filled contour plots in gray scale of te rising bubble by te CHNS system. From left to rigt, t = 0.64, t = 1., t = 1.6, t =, t =.4, t =.8. η = 0.01, λ = , M = 0.1, ν = 0.01, = 0.01, B =

42 Capter Numerical approximations for smectic A liquid crystal flows.1 Introduction Liquid crystal LC is often viewed as te fourt state of te matter besides te gas, liquid and solid. It may flow like a liquid, but its molecules may be oriented in a crystal-like way. Tere are many different types of liquid-crystal pases, wic can be distinguised by teir different optical properties. Termotropic LCs can be distinguised into two main different pases: Nematic and Smectic. In Nematic pases, te rod-saped molecules ave no positional order, but molecules self-align to ave a long-range directional order wit teir long axes rougly parallel. Tus, te molecules are free to flow and teir center of mass positions are randomly distributed as in a liquid, altoug tey still maintain teir long-range directional order. In smectic pases, wic are found at lower temperatures, te well-defined layers form, tat can slide over one anoter in a manner similar to tat of soap. Te smectics are tus positionally ordered along one direction inside te layer. Tere are many different smectic pases, all caracterized by different types and degrees of positional and orientational order cf.[11, 6]. In particular, in Smectic-A pases, molecules are oriented along te normal vector of te layers, wile in Smectic-C pases tey are tilted away from te normal vector of te layer. Te matematical model of liquid crystals can often be derived from an energybased variational formalism energetic variational approaces, leading to well-posed 33

43 nonlinear coupled systems tat satisfy termodynamics-consistent energy dissipation laws. Tis makes it possible to carry out matematical analysis and design numerical scemes wic satisfy a corresponding discrete energy dissipation law. For smectic- A pase liquid crystals, in de Gennes pioneering work [6], te penomenonlogical free energy of smectic A pase is presented by coupling two order parameters wic represent te average direction of molecular alignment, as well as te layer structure, respectively. In [1], te autors modified te de Gennes model by adding a second order gradient term for te smectic order parameter to investigate te nematic to smectic A or smectic C pase transition, and to predict te twist grain boundary pase in ciral smectic liquid crystals. In [4], te autors used te de Gennes energy to study smectic A liquid crystals to simulate te cevron zigzag pattern formed in te presence of an applied magnetic field. In [19], te autors derived te ydrodynamics coupled model for smectic A pase by assuming tat te director field is strictly equal to te gradient of te layer, tus te free energy is reduced to one order parameter. From te numerical point of view, it is specifically desired to design numerical scemes tat could preserve te termo-dynamically consistent dissipation law energy-stable at te discrete level, since te preservation of suc laws is critical for numerical metods to capture te correct long time dynamics. Te noncompliance of energy dissipation laws may lead to spurious numerical solutions if te grid and time step sizes are not carefully controlled. To te best of te autor s knowledge, altoug a variety of te smectic liquid crystal models ad been developed for more tan alf a centry, we notice tat te successful attempts in designing efficient energy stable scemes are very scarce due to te complex nonlinearities. For instances, in [5], te autors present a temporal second order numerical sceme to solve te model of [19]. Te sceme is energy stable, owever, it is nonlinear tus te implementation is complicated and te computational cost migt be ig. In [4], te autors developed a 34

44 temporal first order sceme. However, it does not follow te energy dissipation law even toug te scemes are linear and decoupled. Terefore, te main purpose of tis paper to construct te efficient scemes to solve te de Gennes type smectic A liquid crystal model cf. [6, 4]. We first couple te ydrodynamics to te original de Gennes free energy and derive te wole model based on te variational approac and te Ficks law. To solve te model, te main difficulties rougly include i te coupling between te velocity and director field/layer function troug te convection terms and nonlinear stresses; ii te coupling of te velocity and pressure troug te incompressibility constraint; iii te nonlinear coupling between te director field and layer function. We develop a time discretization sceme wic a is unconditionally stable; b satisfies a discrete energy law; and c leads to linear, decoupled equations to solve at eac time step. Tis is by no means an easy task due to many igly nonlinear terms and te couplings existed in te model. Te rest of te capter is organized as follows. In Section, we present te wole model and present te PDE energy law. In Section 3, we develop te numerical sceme and prove te unconditional stability. In Section 4, we present some numerical experiments to validate te proposed sceme. Finally, some concluding remarks are presented in Section 5.. Te smectic-a liquid crystal fluid flow model and its energy law Te de Gennes free energy of smectic A liquid crystal is described by a unit vector director field d and a complex order parameter ψ, to represent te average direction of molecular alignment and te layer stucture, respectively. Te smectic order parameter is written as ψx = ρxe iqωx,.1 35

45 were ωx is te order parameter to describe te layer structure so tat ω is perpendicular to te layer. Te smectic layer density ρx is te mass density of te layers. Te de Gennes free energy reads as follows, Eψ, d = Ω C ψ iqdψ + K d + g ψ r g χ a Ψ d µ 1 dx. were te order parameters C, K, g, r are all fixed positive constants, µ 1 is te unit vector representing te direction of magnetic field and Ψ is te strengt of te applied field. Ω = L, L d, d. Now we consider te simple case by assuming te density ρx = r/g [4], ten te energy becomes Eψ, d = Ω Cq ω d + K d χ a Ψ d µ 1 dx..3 Let φx = ωx, tus te normalized energy becomes d were 1 φ d Eφ, d = λ + η d τ Ω η d µ 1 dx,.4 x = x d, Ω = 0, l 0,, l = L d, η = γ K d, γ = Cq, λ = dk η, τ = χ aψ d η K..5 Te dimensionless parameter η is in fact te ratio of te layer tickness to te sample tickness and tus η 1. To release te unit vector constraint of d = 1, a nonlinear potential Gb = 1 b 1 wic is a Ginzburg-Landau type penalty term, is added into te free 4ɛ energy to approximate te unit lengt constraint of b [51, 50], were ɛ 1 is a penalization parameter. Tus te modified total free energy wit te ydrodynamics E tot u, φ, d = Ω 1 u dx 1 φ d + λ Ω η + η d 36 + Gd τ d µ 1 dx,.6

46 were u is te fluid velocity field. Assuming a generalized Fick s law tat te mass flux be proportional to te gradient of te cemical potential [7, 4, 54, 53], we can derive te following system: φ t + u φ = M 1 µ φ, µ φ = δe δφ,.7 d t + u d = M µ d, µ d = δe δd,.8 u t + u u σ d + p µ φ φ µ d d = 0,.9 u = 0,.10 were p is te pressure, σ d is te Caugy stress tensor, ν is te viscosity, M 1, M are te relaxation order parameters. Te variational derivatives µ φ and µ d are µ φ = λ 1 φ + d,.11 η µ d = λ η d + gd + 1 η φ + d τd µ 1µ 1,.1 were gd = 1 ɛ d d 1. Following te work in [19], te Caucy stress tensor σ d reads as follows, σ d = µ 1 d T Dudd d + µ 4 Du + µ 5 Dud d + d Dud,.13 were µ i > 0 and Du = 1 u + ut is te strain tensor. In tis paper, we assume µ 1, µ 5 are negligible comparing to µ 4, tus te stress tensor is simplifed to σ d = µ 4 Du..14 For simplicty, we assume te boundary conditions as follows. u Ω = 0, n φ Ω = 0, n d Ω = 0,.15 were n is te outward normal of te boudary. To obtain te dissipation law of te system.7-.10, we take te L inner product of.7 wit δe δe,.8 wit, and.9 wit u, perform te integration by δφ δd 37

47 parts, and add all equalities togeter, we obtain d dt E totu, φ, d = µ 4 Du + M 1 δe Ω δφ + M δe δd dx Numerical sceme Te empasis of our algoritm development is placed on designing numerical scemes tat are not only easy-to-implement, but also satisfy a discrete energy dissipation law. We will design scemes tat in particular can overcome te following difficulties, namely, te coupling of te velocity and pressure troug te incompressibility condition; te stiffness in te director equation associated wit te penalty parameter η; te nonlinear couplings among te fluid equation, te layer equation and te director equation. We construct an energy stable sceme based on a stabilization approac [64]. To tis end, we sall assume tat Gb satisfies te following conditions, i.e., H G x L, x..17 were H G x i,j = G x i x j, i, j = 1,, 3 is te Hessian matrix of Gx. One immediately notes tat tis condition is not satisfied by tis usual double-well potential. However, it is a common practice tat one truncates tis fourt order polynomial G to quadratic growt outside of an interval [ M, M] witout affecting te solution if te maximum norm of te initial condition φ 0 is bounded by M. Terefore, one can cf. [45, 16, 64] consider te truncated double-well potential Gd. We can modify 38

48 tis function outside a ball in {x : dx 1} R 3 of radius 1 as follows. 1 Gd = 4ɛ d 1, d 1, 1 4ɛ d 1, d > 1. Hence, tere exists a postive constant L G suc tat.18 max H Gx L G,.19 x R 3 Wen deriving te energy law.16, we notice tat te nonlinear terms in δetot δφ and δetot δd involve second order derivatives, and it is not convenient to use tem as test functions in numerical approximations, making it difficult to prove te discrete energy dissipation law. To overcome tis difficulty, we first reformulate te system in an alternative form wic is convenient for numerical approximations. Te system reads as follows, φ t + u φ = M 1 µ φ,.0 d t + u d = M µ d,.1 u t + u u σ d + p + φ φ + ḋ d = 0, M 1 M. u = 0,.3 were φ = φ t + u φ and ḋ = d t + u d. To obtain te dissipation law of te system.0-.3, we take te L inner product of.0 wit φ t,.1 wit d t, and. wit u, perform te integration by parts, and add all equalities togeter. We obtain d dt E tot = µ 4 u + 1 φ + 1 ḋ dx..4 Ω M1 M We now fix some notations. For scalar function u, v and vector function u = u 1, u, u 3 and v = v 1, v, v 3, we denote te L inner product as follows. u, v = Ω uvdx, u = u, u, u, v = 39 Ω uv T dx, u = u, u..5

49 Now, we are ready to present our energy stable scemes. Our numerical sceme reads as follows. Given te initial conditions φ 0, ψ 0, u 0 and p 0 = 0, aving computed φ n, ψ n, u n and p n for n > 0, we compute φ n+1, ψ n+1, ũ n+1 u n+1 and p n+1 by wit wit Step 1: Step : 1 M 1 φn+1 = λ η φn+1 d n, φ n+1 n Ω = 0, φ n+1.6 φ n+1 = φn+1 φ n + u n φ n, u n = u n φ n..7 M 1 Sd n+1 d n + 1 ḋ n+1 M d n+1 n Ω = 0, Step 3: = λ η d n+1 gd n + 1 η φn+1 d n+1 + τd n µ 1 µ 1,.8 ḋ n+1 = dn+1 d n ḋn+1 + u n d n, u n = u n d n..9 M ũ n+1 u n + u n ũ n+1 µ 4 ũ n+1 + p n + ũ n+1 Ω = 0. φ n+1 M 1 φ n + ḋn+1 M d n = 0,.30 Step 4: u n+1 ũ n+1 + p n+1 p n = 0, u n+1 = 0, n u n+1 Ω = In te above, S is a stabilizing parameter to be determined. 40

50 We ave te following remarks in order: A pressure-correction sceme [7] is used to decouple te computation of te pressure from tat of te velocity. Te nonlinear term gd mainly takes te form like 1 ɛ d d 1, so te explicit treatment of tis term usually leads to a severe restriction on te time step wen ɛ 1. Tus we introduce in.6 a linear stabilizing" term to improve te stability wile preserving te simplicity. It allows us to treat te nonlinear term explicitly witout suffering from any time step constraint [6, 64, 63]. Note tat tis stabilizing term introduces an extra consistent error of order O in a small region near te interface, but tis error is of te same order as te error introduced by treating it explicitly, so te overall truncation error is essentially of te same order wit or witout te stabilizing term. It is noticable tat te truncation error of te stablizing approac is exactly te same as te convex splitting metod [3]. Inspired by [3, 58, 6], wic deal wit a pase-field model of tree-pase viscous fluids or complex fluids, we introduce two new, explicit, convective velocities u n and u n in te pase equations. u n and u n can be computed directly from.7 and.9, i.e., u n = u n = I + 1 φ n T φ u n n 1 φ n+1 φ n φ n M 1 M1 1 u n 1 d n+1 d n d n M I + M d n T d n, It is easy to get deti + c φ T φ = 1 + c φ φ, tus te above matrix is invertible. Te sceme is a totally decoupled, linear sceme. Indeed,.6,.8 and.30 are respectively decoupled linear elliptic equations for φ n+1, d n+1 and ũ n+1, and.31 can be restated as a Poisson equation for p n+1 p n. 41

51 Terefore, at eac time step, one only needs to solve a sequence of decoupled elliptic equations wic can be solved very efficiently. As we sall sow below, te above sceme is unconditionally energy stable. Teorem.3.1. Under te condition.19, and S ληl, te sceme admits a unique solution satisfying te following discrete energy dissipation law: { E n+1 + pn+1 + ν ũ n+1 + φ n+1 } + ḋn+1 E n + M 1 M pn, were E n = 1 un + λ η d n + Gd n, d n φ n τ η dn µ 1.34 Proof. From te definition of u n and u n in.7 and.9, we can rewrite te momentum equation.30 as follows ũ n+1 u n + u n ũ n+1 µ 4 ũ n+1 + p n = By taking te inner product of.35 wit ũ n+1, and using te identity a b, a = a b + a b,.36 we obtain ũ n+1 u n + ũ n+1 u n + µ 4 ũ n+1 + p n, ũ n+1 = To deal wit te pressure term, we take te inner product of.31 wit p n to derive p n+1 p n p n+1 p n = ũ n+1, p n..38 By taking te inner product of.31 wit u n+1, we obtain u n+1 + u n+1 ũ n+1 = ũ n

52 We also derive from.31 directly tat Combining all identities above, we obtain p n+1 p n = ũ n+1 u n u n+1 u n + ũ n+1 u n + p n+1 p n + ν ũ n+1 = Next, we derive from.7 and.9 tat u n u n u n u n = φ n+1 M 1 φ n,.4 = ḋn+1 M d n..43 By taking te inner product of.4 wit u n, of.43 wit u n, we obtain φ n+1 u n u n + u n u n = φ n, u n M, 1.44 ḋn+1 u n u n + u n u n = d n, u n M..45 Ten, by taking te inner product of.6 wit φ n+1 φ n, we obtain φ n+1 n+1 φ, u n φ n + λ d n, φ n+1 φ n M 1 M 1 η + λ φ n+1 φ n + φ n+1 φ n η = By taking te inner product of.8 wit d n+1 d n, we arrive at S d n+1 d n + ḋn+1 ḋn+1 M M, u n d n d n+1 + λη dn + dn+1 d n + λ d n+1 d n + d n+1 d n η φ n+1, d n+1 d n.47 + ληgd n, d n+1 d n λ η λτ d n µ 1 µ 1, d n+1 d n = 0. 43

53 Combining.41,.44,.45,.46, and.47, we arrive at u n+1 u n + ũ n+1 u n + u n u n + u n u n + p n+1 p n + ν ũ n+1 + φ n+1 + ḋn+1 M 1 M d n+1 + λη dn + dn+1 d n + λ η dn+1 d n + d n+1 d n + λ η φn+1 φ n + φ n+1 φ n.48 + S d n+1 d n + ληgd n, d n+1 d n :Term A + λ η dn, φ n+1 φ n λ η φn+1, d n+1 d n :Term B λτ d n µ 1 µ 1, d n+1 d n :Term C = 0. We deal wit te terms A, B, C as follows. For Term A, we apply te Taylor expansions to obtain g A = ληgd n+1 Gd n, 1 λη 1 ξ, dn+1 d n..49 For Term B, we ave B = λ η = λ η For Term C, we ave d n, φ n+1 φ n + d n+1 d n, φ n+1 d n+1, φ n+1 d n, φ n. C = λτ d n µ 1, d n+1 µ 1 d n µ 1 = λτ d n+1 µ 1 d n µ 1 d n+1 d n µ

54 By combining.48,.49,.50 and.51, we ave u n+1 u n + ũ n+1 u n + u n u n + u n u n + p n+1 p n + ν ũ n+1 + φ n+1 + ḋn+1 M 1 M d n+1 + λη dn + dn+1 d n + ληgd n+1 Gd n, 1 + λ η dn+1 d n + d n+1 d n + λ η φn+1 φ n + φ n+1 φ n λ d n+1, φ n+1 d n, φ n η λτ d n+1 µ 1 d n µ S ληl d n+1 d n + λτ d n+1 d n µ 1 = 0. Finally, we obtain te desired result after dropping some positive terms..4 Numerical Simulations We now present some D numerical experiments to demonstrate te efficiency, stability and accuracy of te propose numerical sceme Te computatational domain is x, y [0, l] [0, ]. For x-axis, we set te periodic boundary condition, and for y axis, we set Neumann or Diricelet boundary conditions. We adopt te second order central finite difference metod to discretize te space. Te magnectic field is always set as µ 1 = 0, 1. If not explicitly specified, te default values of order parameters are given as follows, l =, ɛ = 0.0, η = 0.0, M 1 = 0.08, M =, λ =.5, τ =

55 Figure.1: Te L errors of te layer funciton φ, te director field d = d 1, d, te velocity u = u, v and pressure p. Te slopes sow tat te sceme is asymptotically first-order accurate in time..4.1 Accuracy test We first test te convergence rate of sceme We set te following initial conditions as dt = 0 = sinπxcosπy, cosπxcosπy, φt = 0 = cosπy,.54 ut = 0, pt = 0 = 0, 0. Te boundary conditions are Neumann type along y-axis cf..15. We use grid points to discretize te space, and perform te mes refinement test for time. We coose te numerical solution wit te time step size = as te bencmark solution approximate exact solution for computing errors. Figure.1 plots te L errors for various time step sizes. We observe tat te sceme is asymptotically first-order accurate in time for all variables as expected. 46

56 .4. Cevron pattern induced by te magnetic force We now consider te effects from te magnetic force for te no flow case u = 0. Initially, a smectic A liquid crystal is confined between two flat parallel plates and uniformly aligned in a way tat te smectic layers are parallel to te bounding plates and te directors are aligned omeotropically, tat is, perpendicular to te smectic layers. A magnetic field is applied in te direction parallel to te smectic layers, wic induce te layer undulation cevron pattern penomena. Te initial conditions read as follows. dt = 0 = 0, randx, y, randx, y, φt = 0 = y,.55 were te randx, y is te small perturbation tat is te random number in [ 1, 1] and as zero mean. We set te Diriclet type boundary condition for φ and d as follows, d y=±1 = 0, 1, φ y=1 = 1, φ y= 1 = We take = to obtain better accuracy. Fig.. sows te snapsots of te layer function φ at t = 0, 0., 0.4 and 0.8. Initially at t = 0, te layer function take te linear profile along te y axis. Wen time evolves, we observe some undulations appear at t = 0.. Te layer function quickly reaces te steady solution at t = 0.8 wit te saw toot sape. Tis undulation penomenon is called te Helfric-Hurault effect cf. [34, 38]. Fig..3 sows te snapsots of te directior field d. Te numerical solution presents similar features to tose obtained in [4]. We also plot te energy dissipative curve in Fig..4, wic confirms tat our algoritm is energy stable. 47

57 Figure.: Snapsots of te layer function φ are taken at t = 0, 0., 0.4 and 0.8 for Example Cevron pattern induced by magnetic force and sear flow We now impose te sear flow on te top and bottom plates to see ow te flow affects te undulation. Te initial and boundary conditions of φ and d are same as te example.4.. For velocity and pressure, te initial and boundary conditions are: ut = 0 = 10y 1, 0, pt = 0 = 0 u y=1 = 10, 0, u y= 1 = 10, Fig..5 sow te snapsots of te layer function φ at t = 0, 0.3, 0.4, 0.5, 0.6 and 0.8. Wen time evolves, te layer undulations still appear but te symmetry is largely disturbed by te sear flow. Fig..6 sows te snapsots of te directior field d. We also plot te first component of te velocity field u = u, v in Fig..7, were te linear profile is deformed to sow nonlinearility. 48

58 Figure.3: Snapsots of te director field d are taken at t = 0, 0., 0.4 and 0.8 for Example.4.. Figure.4: Time evolution of te free energy functional of Example

59 Figure.5: Snapsots of te layer function φ are taken at t = 0, 0.3, 0.4, 0.5, 0.6 and 0.8 for Example