c 2009 Society for Industrial and Applied Mathematics

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1 SIAM J NUMER ANAL Vol 47 No 3 pp c 009 Society for Industrial and Applied Mathematics AN ENERGY-STABLE AND CONVERGENT FINITE-DIFFERENCE SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION S M WISE C WANG AND J S LOWENGRUB Abstract We present an unconditionally energy stable finite-difference scheme for the phase field crystal equation The method is based on a convex splitting of a discrete energy and is semiimplicit The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size We present local-in-time error estimates that ensure the convergence of the scheme While this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related Swift Hohenberg equation as well Key words phase field crystal finite-difference methods stability nonlinear partial differential equations AMS subject classifications 35G5 65M06 65M1 DOI / Introduction The phase field crystal PFC model was recently proposed in [7] as a new approach to simulating crystals at the atomic scale in space but on a coarse-graineddiffusive time scale The PFC model accounts for the periodic structure of a crystal lattice through a free energy functional of Swift Hohenberg SH type [17] that is minimized by periodic functions The model can account for elastic and plastic deformations of the lattice dislocations grain boundaries and many other observable phenomena See for example the recent review [15] that describes the variety of microstructures that can be modeled using the PFC approach The idea is that the phase variable describes a coarse-grained temporal average of the number density of atoms and the approach can be related to dynamic density functional theory [ 13] Consequently this method represents a significant advantage over other atomistic methods such as molecular dynamics where the time steps are constrained by atomic-vibration time scales Presently we introduce the PFC equation and the closely related SH equation and give an idea of the numerical schemes detailed in section 3 We also point out some of the previous theoretical and numerical work surrounding the SH and PFC equations 11 Phase field crystal equation Herein we consider a dimensionless energy of the form [7 17] { 1 11 Eφ = Ω 4 φ4 + 1 ɛ φ φ + 1 } Δφ dx Received by the editors October ; accepted for publication in revised form April ; published electronically June The research of the first and third authors was partially supported by the National Science Foundation Division of Mathematical Sciences and Division of Materials Research Corresponding author Mathematics Department The University of Tennessee Knoxville TN swise@mathutkedu Mathematics Department The University of Massachusetts North Dartmouth MA 0747 cwang1@umassdedu Mathematics Department The University of California Irvine CA 9697 lowengrb@mathuci edu 69

2 70 S M WISE C WANG AND J S LOWENGRUB where Ω R D whered = or 3 φ :Ω R is the density field ɛ is a constant and and Δ are the gradient and Laplacian operators respectively Suppose that Ω=0L x 0L y andthatφ and Δφ are periodic on Ω We consider two types of gradient dynamics on Ω: i nonconserved dynamics SH [17] 1 t φ = Mμ where M>0 is a mobility μ is the chemical potential defined as 13 μ := δ φ E = φ 3 +1 ɛ φ +Δφ +Δ φ and δ φ E denotes the variational derivative with respect to φ; and ii conserved dynamics PFC [7] 14 t φ = Mφ μ where Mφ > 0 is a mobility and where we assume that μ is periodic on Ω Because the dynamical equations are of gradient type it is easy to see that the energy 11 is nonincreasing in time along the solution trajectories of either 1 or 14 Equation 14 is a mass conservation equation where the flux is proportional to the gradient of the chemical potential This along with the periodic boundary conditions ensures that Ω tφdx =0 Equation 1 is referred to as the Swift Hohenberg SH equation [17] and is fourth-order in space Equation 14 is the phase field crystal PFC equation and is a sixth-order equation In this paper we wish to describe a practical computational scheme for solving the PFC equation primarily though the theory will be applicable to the SH equation as well We wish to point out that while periodic boundary conditions are assumed herein the theory and numerical analysis to follow also hold for homogeneous Neumann boundary conditions as well as for mixed periodic-homogeneous Neumann boundary conditions 1 Discrete-time continuous-space schemes To motivate the fully discrete methods that are to come we first present related schemes at the discrete-time continuous-space level Here the discussion will be formal However in the next sections we develop a rigorous theory in the finite-dimensional framework The fundamental observation is that the energy E admits a not necessarily unique splitting into purely convex and concave energies that is E = E c E e where E c and E e are convex though not necessarily strictly convex Assuming that ɛ<1 the canonical splitting is { 1 15 E c = 4 φ4 + 1 ɛ φ + 1 } Δφ dx E e = φ dx Ω We note that the splitting is easy to modify in the case that ɛ>1 This observation about convex splitting was first exploited by Eyre [8] to craft energy-stable numerical schemes We borrow the notation E c and E e from [8] where c refers to the contractive part of the energy and e refers to the expansive part of the energy The schemes we propose are of the type proposed by Eyre but the theory to establish solvability and energy stability here is more direct and more complete We also point out that Eyre did not specifically consider the SH or PFC equations though his framework was Ω

3 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 71 sufficiently general to do so In particular we show that some of the assumptions made in [8] are unnecessary to prove energy or gradient stability More importantly we also establish solvability convergence and an error estimate for the numerical method for the conserved dynamics case PFC which were not pursued in [8] Our approach is based upon the following simple but fundamental estimate that was not utilized in [8] We will be primarily interested in its finite-dimensional counterpart in section 3 Theorem 11 Suppose that Ω=0L x 0L y and φ ψ :Ω R are periodic and sufficiently regular and suppose that Δφ and Δψ are also periodic Consider the canonical convex splitting of the energy E in 11 into E = E c E e given in 15 Then 16 Eφ Eψ δ φ E c φ δ φ E e ψφ ψ L where δ φ denotes the variational derivative Proof Let E c φ = Ω e cφ x φ y φ Δφ dx Since e c φ where φ = φ 1 φ φ 3 φ 4 is convex in all of its arguments we have the equivalent statement 17 e c ψ e c φ φ e c φ ψ φ for any φ ψ R 4 Setting φ =φ x φ y φ Δφ andψ =ψ x ψ y ψ Δψ and integrating 17 we get { E c ψ E c φ φ e c φ x φ y φ Δφψ φ 18 Ω + xφe c φ x φ y φ Δφ x ψ x φ + yφe c φ x φ y φ Δφ y ψ y φ } + Δφ e c φ x φ y φ ΔφΔψ Δφ dx Integration by parts leads to the inequality [6 Chap ] 19 E c ψ E c φ δ φ E c φψ φ L By a similar analysis on E e but reversing the roles of φ and ψ wehave 110 E e φ E e ψ δ φ E e ψφ ψ L Adding 19 and 110 yields Eψ Eφ =E c ψ E e ψ E c φ E e φ 111 δ φ E c φψ φ L +δ φ E e ψφ ψ L =δ φ E c φ δ φ E e ψψ φ L For nonconserved dynamics we propose the semi-implicit scheme 11 φ k+1 φ k = sm μ μ φ k+1 φ k := δ φ E c φ k+1 δ φ E e φ k where s>0 is the time step size Here δ φ E c = φ 3 +1 ɛφ+δ φandδ φ E e = Δφ Setting φ = φ k+1 and ψ = φ k in 16 and using 11 we have 113 Eφ k+1 Eφ k δ φ E c φ k+1 δ φ E e φ k φ k+1 φ k L = μ sm μ L = sm μ L 0

4 7 S M WISE C WANG AND J S LOWENGRUB Hence the energy is nonincreasing in time regardless of time step size: 114 Eφ k+1 Eφ k In this case we say that the scheme is unconditionally energy stable Note also that the scheme 11 is unconditionally solvable based on the convex structure of the equations but is generally nonlinear For conserved dynamics we propose the semi-implicit scheme 115 φ k+1 φ k = s Mφ k μ μ φ k+1 φ k := δ φ E c φ k+1 δ φ E e φ k Again we set φ = φ k+1 and ψ = φ k in 16 and find Eφ k+1 Eφ k δ φ E c φ k+1 δ φ E e φ k φ k+1 φ k L 116 = s μ Mφ k μ L = s μ Mφ k μ L 0 where we have dropped the boundary terms coming from integration-by-parts using the periodicity of μ As before the energy is always nonincreasing in time 117 Eφ k+1 Eφ k and thus the scheme is unconditionally energy stable Moreover we expect that it is unconditionally solvable based on convexity arguments For both schemes the global truncation error is first-order in time We reiterate that schemes of this type were proposed by Eyre [8] primarily for solving the Cahn Hilliard [4 3] and Allen Cahn [1] equations but our analysis here is both more direct and more complete We also note that the methodology herein is quite general and applicable to a broad class of gradient problems There has been some limited work developing stable schemes for the PFC equation recently Cheng and Warren [5] and Mellenthin Karma and Plapp [14] describe linear spectral schemes for solving the PFC equation on periodic domains Cheng and Warren s scheme is similar to one for the Cahn Hilliard equation analyzed in [18] and depends upon a Fourier stability analysis of a linearized PFC equation where in particular the nonlinear term φ 3 in the chemical potential is treated explicitly Mellenthin et al use an integrating factor approach Neither method is a convex splitting scheme The finite element method of Backofen Rätz and Voigt [] employs what is essentially a standard backward Euler scheme but where the nonlinear term φ 3 in the chemical potential is linearized via φ k+1 3 3φ k φ k+1 φ k 3 No stability analysis is given though they claim that relatively large step sizes can be achieved None of the preceding works claim to propose rigorously energy-stable methods The schemes mentioned above are linear and consequently conditions on solvability are somewhat more accessible than in the nonlinear case We point out that because Backofen et al keep the Δφ term implicit in the treatment of the chemical potential their scheme is not expected to be unconditionally uniquely solvable We mention that an alternate approach to the nonlinear splitting scheme proposed here is a linear splitting scheme as was suggested by Eyre [8] for the Cahn Hilliard equation and by Xu and Tang [0] for a bistable epitaxial thin film equation This would involve a splitting of the chemical potential such as μ = A φ k+1 φ k B Δφ k+1 Δφ k +Δ φ k+1 + φ k 3 +1 ɛ φ k +Δφ k

5 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 73 where the splitting parameters A B 0 must be determined in order to ensure stability Unconditional unique solvability is guaranteed thanks to the linearity and positivity of the respective terms It is expected that as in [0] the energy will be nonincreasing in time provided A and B are sufficiently large but that such A and B will depend on the unknown φ k+1 [0 inequality 14] In what follows we demonstrate that the convex-splitting framework outlined in this section has an analogue in the finite-dimensional setting using difference operators In section we describe the two-dimensional finite-difference operators summation-by-parts formulae discrete norms and discrete estimates that will facilitate the definition and analysis of our schemes In section 3 we present the main results of our analysis including the unique solvability discrete-energy stability and convergence of our schemes We give some concluding remarks and suggest some future work in section 4 Discretization of two-dimensional space Our primary goal in this section is to define finite-difference operators and summation-by-parts formulae in two space dimensions needed to define and analyze the schemes We begin by establishing some machinery in one space dimension 1 One-dimensional difference operators and summation-by-parts formulae Suppose Ω = 0L Let h = L/n wheren Z + We define the function x r = xr :=r 1 / h wherer takes on integer and half-integer values and the following three sets: E n = {x i+ 1/ i =0n} C n = {x i i =1n} and C n = {x i i =0n+1} We define by E n the space of functions whose domains equal E n The functions of E n are called edge-centered functions and we reserve the symbols f and g to denote them In component form these functions are identified via f i+ 1/ := fx i+ 1/ fori =0nByC n we denote the vector space of functions whose domains are equal to C n andbyc n we denote the space of functions whose domains equal C n The functions of C n and C n are called cell-centered functions and we use the Greek symbols φ ψ andζ to denote them In component form these functions are identified via φ i := φx i where i =1n if φ C n andi =0n+1 if φ C n Let φ and ψ be cell-centered and let f and g be edge-centered We define the respective inner-products on C n and E n : 1 φ ψ = n φ i ψ i [f g] = 1 i=1 n fi 1/g i 1/ + f i+ 1/g i+ 1/ i=1 The edge-to-center difference operator d : E n C n ; the center-to-edge average and difference operators respectively A D : C n E n ; and the discrete Laplacian operator Δ h : C n C n are defined componentwise as 3 4 df i = 1 fi+ 1/ f i 1/ i=1n h Aφ i+ 1/ = 1 φ i + φ i+1 Dφ i+ 1/ = 1 h φ i+1 φ i i=0n Δ h φ i = d Dφ i = 1 h φ i 1 φ i + φ i+1 i=1n The following summation-by-parts formulae are established straightforwardly

6 74 S M WISE C WANG AND J S LOWENGRUB Proposition 1 Let φ ψ C n and f E n Then h [Dφ f] = h φ df Aφ1/f1/ + Aφ n+ 1/f n+ 1/ h [Dφ Dψ] = h φ Δ h ψ Aφ1/Dψ1/ + Aφ n+ 1/Dψ n+ 1/ h φ Δ h ψ=h Δ h φ ψ+aφ n+ 1/Dψ n+ 1/ Dφ n+ 1/Aψ n+ 1/ Aφ1/Dψ1/ + Dφ1/Aψ1/ Two-dimensional difference operators and summation-by-parts formulae Let Ω = 0L x 0L y where L x = m h and L y = n h whereh>0 As in the one-dimensional case we define E m C m andc m with respect to 0L x and E n C n andc n with respect to 0L y Define the function spaces C m n = {φ : C m C n R} C m n = {φ : C m C n R} C m n = {φ : C m C n R} C m n = {φ : C m C n R} E ew m n = {f : E m C n R} E ns m n = {f : C m E n R} The functions of C m n C m n C m n andc m n are called cell-centered functions andweusethegreeksymbolsφ ψ andζ to denote them In component form these functions are identified via φ ij : φx i y j where like x r y r =r 1 / h The functions of Em n ew and Ens m n are called east-west edge-centered functions and northsouth edge-centered functions respectively We use the symbols f and g to denote these functions In component form east-west edge-centered functions are identified via f i+ 1/j := fx i+ 1/y j ; north-south edge-centered functions are identified via f ij+ 1/ := fx i x j+ 1/ We define the following weighted inner-products: φ ψ = [f g] ew = 1 [f g] ns = 1 m i=1 j=1 m n φ ij ψ ij φ ψ C m n C m n C m n C m n n i=1 j=1 m n i=1 j=1 fi+ 1/jg i+ 1/j + f i 1/jg i 1/j f g E ew m n fij+ 1/g ij+ 1/ + f ij 1/g ij 1/ f g E ns m n The edge-to-center differences d x : Em n ew C m n and d y : Em n ns C m n ; the center-to-edge averages and differences A x D x : C m n Em n ew and A y D y : C m n Em n ew ; and the two-dimensional discrete Laplacian Δ h : C m n C m n are defined componentwise via d x f ij = 1 fi+ 1/j f i 1/j dy f ij = 1 fij+ 1/ f ij 1/ i=1m j=1n h h A x φ i+ 1/j = 1 φ ij + φ i+1j D x φ i+ 1/j = 1 h φ i+1j φ ij i=0m j=1n 16 A y φ ij+ 1/ = 1 φ ij + φ ij+1 D y φ ij+ 1/ = 1 h φ ij+1 φ ij i=1m j=0n 17 Δhψij = dxdxψij + dydyψij i=1m j=1n The following formulae follow directly

7 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 75 Proposition If φ ψ C m n f Em n ew andg Ens m n then h [D x φ f] ew = h φ d x f h 18 A x φ1/ h [D y φ g] = ns h φ d y g f1/ + h Ax φ m+ 1/ fm+ 1/ 19 h A y φ 1/ g 1/ + h Ay φ n+ 1/ g n+ 1/ where the indicates the direction along which the one-dimensional inner-product acts and h [D x φ D x ψ] ew + h [D y φ D y ψ] ns = h φ Δ h ψ h A x φ1/ Dx ψ1/ + h Ax φ m+ 1/ Dx ψ m+ 1/ h 0 A y φ 1/ Dy ψ 1/ + h Ay φ n+ 1/ Dy ψ n+ 1/ h φ Δ h ψ=h Δ h φ ψ 1 + h A x φ m+ 1/ h A x φ1/ Dx ψ1/ + h A y φ n+ 1/ h A y φ 1/ Dy ψ 1/ Dx ψ m+ 1/ h Dx φ m+ 1/ Ax ψ m+ 1/ + h Dx φ1/ Ax ψ1/ Dy ψ n+ 1/ h Dy φ n+ 1/ Ay ψ n+ 1/ + h Dy φ 1/ Ay ψ 1/ 3 Periodic boundary conditions In this paper we use periodic grid functions Specifically we shall say the cell-centered function φ C m n is periodic if and only if 3 φ m+1j = φ 1j φ 0j = φ mj j =1n φ in+1 = φ i1 φ i0 = φ in i =0m+1 For such functions the center-to-edge averages and differences are periodic For example if φ C m n is periodic then A x φ m+ 1/j = A x φ1/j and D x φ m+ 1/j = D x φ1/j for all j =0 1n+ 1 We also note that the results for periodic functions that are to follow will also hold in possibly slightly modified forms when the boundary conditions are taken to be homogeneous Neumann 4 5 φ m+1j = φ mj φ 0j = φ 1j j =1n φ in+1 = φ in φ i0 = φ i1 i =0m+1 or a mixture of homogeneous Neumann and periodic boundary conditions 4 Norms We define the following norms for cell-centered functions If φ C m n then φ := h φ φ φ 4 := h φ 4 1 and φ := max 1 i m φ ij We define h φ whereφ C m n tomean 1 j n 6 h φ := h [D x φ D x φ] ew + h [D y φ D y φ] ns We will use the following discrete Sobolev-type norms for grid functions φ C m n : φ 0 := φ and 7 φ 1 := φ + hφ φ := φ + hφ + Δ hφ

8 76 S M WISE C WANG AND J S LOWENGRUB 5 Discrete Sobolev inequalities Here we prove a discrete Sobolev-type inequality for two-dimensional grid functions that will be needed to prove pointwise stability The primary result is Lemma 5 The two preliminary lemmas Lemmas 3 and 4 are similar to results proved in [1 sec 86] and we skip their proofs Lemma 3 Suppose that φ C n Then for any i {1 n} 8 φ i h φ φ+lh [Dφ Dφ] L Lemma 4 Suppose that φ C m n Then for any i {1 m} and any j {1 n} 9 φ ij 4 h φ φ+ 4L x h [D x φ D x φ] L x L y L ew + 4L y h [D y φ D y φ] y L ns x m n +4L x L y h wi m Dy wn j D x φ i + 1 /j + 1 / i =0 j =0 where wk n =1if k {1 n 1} and wn k =1if k {0n} Lemma 5 Suppose that φ C m n is periodic Then for any i {1 m} and any j {1 n} { 30 φ ij 1 4max L x L y L } xl y φ L x L y L y L x Hence φ C φ wherec depends only upon L x and L y Proof Using the previous lemma this result will follow if we can show m n 31 S := h wi m Dy wn j D x φ i + 1 /j / Δ hφ i =0 j =0 By definition of the edge inner-product [ ] 3 S = h m ] wi [D m h y D x φ i + 1 / D y D x φ i + 1 / i =0 Using summation-by-parts Proposition 1 and dropping the boundary terms due to periodicity yields 33 S = h m wi Δ m h y h D xφ i + 1 / D x φ i + 1 / i =0 where Δ h := d D Note the commutation of operators: Δ y h D x = D x Δ y h Thus 34 S = h = h = h m wi D m h x Δ y h φ i + 1 / D x φ i + 1 / i =0 m wi m h n D x Δ y h φ i + 1 /j D x φ i + 1 /j i =0 n h j =1 j =1 [ D x Δ y h φ j Dx φ j ]

9 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 77 Again using summation-by-parts Proposition 1 and dropping the boundary terms due to periodicity yields S = h n h Δ y h φ j Δx h φ j =h Δ y h φ Δx h φ j =1 h Δ y h φ Δx h φ+h Δx h φ Δx h φ+h Δy h φ Δy h φ 35 = h Δx h φ +Δy h φ Δx h φ +Δy h φ=h Δ hφ Δ h φ= 1 Δ hφ 3 Main results 31 A convex splitting of the discrete energy We begin by defining a fully discrete energy that is consistent with the continuous space energy 11 In particular define the discrete energy F : C m n R to be 31 F φ := 1 4 φ ɛ φ hφ + 1 Δ hφ Lemma 31 existence of a convex splitting Suppose that φ C m n is periodic and that Δ h φ C m n is also periodic Define the energies 3 F c φ := 1 4 φ ɛ φ + 1 Δ hφ F eφ := h φ Then the gradients of the respective energies are δ φ F c = φ 3 +1 ɛφ +Δ h φ and δ φ F e = Δ h φandf c and F e are convex provided ɛ<1 Hence F as defined in 31 admits the convex splitting F = F c F e Proof Suppose that ψ C m n is periodic such that Δ h ψ C m n is also periodic Calculating the discrete variation of F c and using summation-by-parts Proposition eq 1 shows 33 df c ds φ + sψ = h φ 3 +1 ɛφ +Δ h φ ψ s=0 and the gradient formula follows A calculation of the second variation reveals d F c 34 φ + sψ ds = h 3φ ψ +1 ɛψ +Δ h ψ 1 0 s=0 which proves that F c is convex In fact it is strictly convex For F e calculating the variation and using summation-by-parts Proposition eq 0 gives df e 35 φ + sψ ds = h Δ h φ ψ s=0 which yields the gradient Consider the second variation of the energy: 36 d F D ds φ + sψ =h [D x ψ D x ψ] ew +h [D y ψ D y ψ] ns Hence d F e ds φ + sψ s=0 0 which implies that F e is convex

10 78 S M WISE C WANG AND J S LOWENGRUB We now describe the fully discrete schemes in detail chemical potential μ C m n to be Define the cell-centered μ φ k+1 φ k := δ φ F c φ k+1 δ φ F e φ k 37 = φ k ɛφ k+1 +Δ h φ k +Δ h κ where κ := Δ h φ k+1 The semi-implicit scheme for nonconserved dynamics ie the SH equation is as follows: given φ k C m n periodic find φ k+1 κ C m n periodic such that 38 φ k+1 φ k = sm μ M > 0 The scheme for conserved dynamics ie the PFC equation is the following: given φ k C m n periodic find φ k+1 μ κ C m n periodic such that { 39 φ k+1 φ k = s d x M Ax φ k D x μ + d y M Ay φ k D y μ } Mφ > 0 3 Unconditional unique solvability We now show how the convexity translates into solvability for the schemes We first need two lemmas that will help prove solvability for the mass-conserving scheme 39 Lemma 3 Suppose the edge-centered functions M ew Em n ew and M ns Em n ns are positive For any φ C m n there exists a unique ψ C m n that solves 310 Lψ := d x M ew D x ψ d y M ns D y ψ=φ 1 m n φ ψ is periodic and ψ 1 =0 Proof Write ˆφ = φ φ 1 /m n sothat ˆφ has mean zero ie ˆφ 1 =0 Then 310 is Lψ = ˆφ Assuming that ψ is a periodic solution of the last equation then the necessity of ˆφ 1 = 0 follows from ˆφ 1 = d x M ew D x ψ 1 d y M ns D y ψ 1 31 =[M ew D x ψ D x 1] ew +[M ns D y ψ D y 1] ns =0 We now show that L is symmetric and positive definite restricted to the subspace of mean-zero functions Symmetry follows from the next calculation Suppose ψ 1 ψ C m n are periodic Then using summation-by-parts ψ 1 Lψ = ψ 1 d x M ew D x ψ d y M ns D y ψ =[D x ψ 1 M ew D x ψ ] ew +[D y ψ 1 M ns D y ψ ] ns 313 =[M ew D x ψ 1 D x ψ ] ew +[M ns D y ψ 1 D y ψ ] ns = d x M ew D x ψ 1 d y M ns D y ψ 1 ψ =Lψ 1 ψ where the boundary terms vanish because of periodicity Setting ψ = ψ 1 = ψ 314 ψ Lψ = [M ew D x ψ D x ψ] ew +[M ns D y ψ D y ψ] ns 0 from which we get that L is positive semidefinite Equality in the last line is achieved only when D x ψ =0andD y ψ =0atevery edge-centered point However the only

11 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 79 way that D x ψ =0andD y ψ =0atevery edge point is if ψ is a constant function as may be easily shown With the restriction that ψ has mean zero ψ 1 = 0this constant must be zero which proves that L is positive definite Suppose the edge-centered functions M ew Em n ew and M ns Em n ns are positive Define the vector space 315 H := {φ C m n φ 1 =0} and equip this space with the bilinear form 316 φ 1 φ HL := [M ew D x ψ 1 D x ψ ] ew +[M ns D y ψ 1 D y ψ ] ns for any φ 1 φ H whereψ i C m n is the unique solution to 317 Lψ i = d x M ew D x ψ i d y M ns D y ψ i =φ i ψ i periodic ψ i 1 =0 If φ C m n thenwetakeφ H to mean that the restriction of φ to C m C n is in H Lemma 33 φ 1 φ HL is an inner-product on the space H Moreover 318 φ 1 φ HL = φ 1 L 1 φ = L 1 φ 1 φ Proof Since L is symmetric and positive definite SPD it has an inverse L 1 and the inverse must also be SPD The equality of the three bilinear forms follows from summation-by-parts and φ 1 L 1 φ is easily seen to be an inner-product Theorem 34 unique solvability The nonconserved scheme 38 and the conserved scheme 39 are uniquely solvable for any time step size s>0 Moreover the scheme 39 is discretely mass conserving ie φ k+1 φ k 1 =0 Proof Unique solvability of the nonconserved scheme results from the fact that the functional 319 Gφ = h φ φ+smf cφ h φ φ k + smδ φ F e φ k is strictly convex with respect to φ and its minimization is equivalent to solving 38 Discrete mass conservation of 39 follows from φ k+1 φ k 1 = s dx M Ax φ k D x μ + d y M Ay φ k D y μ 1 30 = s [ M A x φ k D x μ Dx 1 ] ew s [ M A y φ k D y μ Dy 1 ] ns =0 Thus if 39 has a solution φ k+1 then by necessity it must be that φ k+1 1 = φ k 1 ; ie φ k+1 and φ k have equal means Now without loss of generality we may suppose that φ k H whereh is the subspace of mean-zero functions in C m n defined above The appropriate space for solutions of 39 must necessarily be H Now consider the following functional on H: 31 Gφ = h φ φ HL h φ φ k HL + F cφ h φ δφ F e φ k where the H inner-product is defined as 3 φ 1 φ HL := [ sma x φ k ] D x ψ Dx 1 ψ ew + [ sma y φ k D y ψ Dy 1 ψ ]ns

12 80 S M WISE C WANG AND J S LOWENGRUB and ψ i C m n is the unique solution by Lemma 3 to 33 Lψ i = sd x MAx φ k D x ψ i sdy MAy φ k D y ψ i = φi such that ψ i is periodic and is mean-zero ψ i 1 = 0 By Lemma 33 G is equivalent to 34 Gφ = h L 1 φ φ h L 1 φ φ k + F c φ h φ δφ F e φ k and it is clear that this functional is strictly convex if φ is restricted to H Moreover it may be shown that φ k+1 H is the unique minimizer of G if and only if it solves 35 δ φ Gφ k+1 =L 1 φ k+1 φ k +δ φ F c φ k+1 δf e φ k C =0 where C is a constant Since L 1 φ k+1 φ k has zero mean it must be that δ φ F c φ k+1 δf e φ k C has zero mean in which case 36 C = 1 δφ F c φ k+1 δf e φ k 1 m n Applying the invertible operator L to 35 we have 37 φ k+1 φ k = s { d x M Ax φ k D x μ + d y M Ay φ k D y μ } + LC But of course LC = 0 Hence minimizing the strictly convex functional 34 is equivalent to solving Unconditional stability The following is a discrete version of Theorem 11 Theorem 35 energy decay estimate Suppose that φ k+1 φ k C m n are periodic and that Δ h φ k+1 C m n is also periodic Assume that the discrete energy F is as given in 31 and take the convex splitting F = F c F e in Lemma 31 Then 38 F φ k+1 F φ k h δ φ F c φ k+1 δ φ F e φ k φ k+1 φ k Proof We have by convexity F c φ k F c φ k+1 h δ φ F c φ k+1 φ k φ k+1 F e φ k+1 F e φ k h δ φ F e φ k φ k+1 φ k Adding the inequalities and multiplying by 1 gives the result The energy decay estimate readily yields the energy stability of the schemes Theorem 36 energy stability Both the nonconserved 38 and conserved 39 schemes are unconditionally strongly energy stable meaning that for any time step size s>0 331 F φ k+1 F φ k Proof For the nonconserved case using the energy decay estimate we have F φ k+1 F φ k h δ φ F c φ k+1 δ φ F e φ k φ k+1 φ k = h μ sm μ = smh μ μ 0

13 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 81 For the conserved case using the energy decay estimate and summation-by-parts we have F φ k+1 F φ k h δ φ F c φ k+1 δ φ F e φ k φ k+1 φ k = h μ s { d x M Ax φ k D x μ + d y M Ay φ k D y μ } = sh {[ D x μ M Ax φ k D ] x μ ew + [ D y μ M Ay φ k D y μ ] } ns 0 Using an idea from [9] we now show that energy stability leads to the uniform boundedness of the discrete solutions First we need two lemmas Lemma 37 Suppose that φ C m n is periodic Then 33 F φ C φ L xl y 4 where C>0 is independent of h Proof First note that using α 1 0 we have φ φ L xl y 4 We also have the following from summation-by-parts and Cauchy s inequality: 334 h φ = h φ Δ h φ 1 β φ + β Δ hφ valid for any β>0 Hence 335 h φ + β Δ hφ 1 β φ Set a =1 ɛ>0 Putting the above estimates together we have 336 F φ = 1 4 φ a φ hφ + 1 Δ hφ 1 + a 1 φ 1 β + β Δ h φ L xl y 4 We choose β such that 1 /+ a / 1 /β > 0and 1 / β / > 0 The last set of inequalities is equivalent to 1 + a> 1 /β and 1 >β which always has a solution in the quadrant where β a > 0 In particular we take β = β 0 := 1 / 1+a Thus 337 F φ K φ + Δ hφ L xl y 4 where K =min 1 / + a / 1 /β 0 1 / β0 / > 0 Using inequality 334 again but with β =1wehave 338 F φ K 3 and the desired result follows φ + hφ + Δ hφ L xl y 4

14 8 S M WISE C WANG AND J S LOWENGRUB Lemma 38 Suppose that φ C m n is periodic Then 339 F φ C φ L xl y 4 where C>0 is independent of h Proof This follows from the discrete Sobolev inequality of Lemma 5 Finally we can prove the uniform pointwise boundedness of the discrete solution of either the SH or PFC scheme Theorem 39 Let Φx y be a smooth periodic function on Ω=0L x 0 L y and φ 0 ij := Φx iy j and suppose E is the continuous energy 11 Letφ k ij C m n be the kth periodic solution of either of the schemes 38 or 39 Then 340 φ k C 1 EΦ + C L x L y where C 1 C > 0 and neither C 1 nor C depend on either s or h Proof From the energy stability Theorem 36 and the previous Lemma 38 we have 341 F φ 0 F φ k C φ k L xl y 4 where C>0doesnotdepend upon h It is straightforward to show eg [9 Cor 1] that 34 F φ 0 =EΦ + τ τ Mh ML x L y where τ is an approximation error and M is some positive constant that does not depend upon h Then 343 ML x L y + EΦ C φ k L xl y 4 and the result follows 34 Error estimate for the PFC equation We conclude this section with a local-in-time error estimate for the PFC equation A similar result for the SH equation may be obtained without difficulty For the PFC scheme we will need the following estimate showing control of the backward diffusion term Lemma 310 Suppose that φ C m n is periodic and that Δ h φ C m n is also periodic Then 344 Δ h φ 1 3α φ + α 3 h Δ h φ valid for arbitrary α>0 Proof Using summation-by-parts Proposition eq 0 and Cauchy s inequality Δ h φ = h Δ h φ Δ h φ=h [ D x φ D x Δ h φ] ew + h [ D y φ D y Δ h φ] ns 345 h α [D xφ D x φ] ew + h α [D x Δ h φ D x Δ h φ] ew + h α [D yφ D y φ] ns + h α [D y Δ h φ D y Δ h φ] ns = 1 α hφ + α h Δ h φ

15 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 83 Again using summation-by-parts and Cauchy s inequality we have the estimate 346 h φ = h [D x φ D x φ] ew + h [D y φ D y φ] ns = h φ Δ h φ h α φ φ+h α Δ hφ Δ h φ= 1 α φ + α Δ hφ Putting the two estimates together we find 347 Equivalently Δ h φ 1 α 1 α φ + α Δ hφ + α h Δ h φ = 1 4α φ Δ hφ + α h Δ h φ Δ hφ 1 4α φ + α h Δ h φ and the desired result follows The existence and uniqueness of a smooth periodic solution to the PFC equation may be established using standard techniques See Remark 31 In the following pages we denote this PDE solution by Φ and in the next theorem we establish an error estimate for the fully discrete approximation to Φ Theorem 311 error estimate For simplicity assume that Mφ =1in 14 and let Ω=0L x 0 L y Given smooth periodic initial data Φx y t =0 suppose the unique smooth periodic solution for the PFC equation 14 is given by Φx y t on Ω for 0 <t T forsomet < Define Φ k ij := Φx iy j ks and e k ij := Φk ij φk ij whereφk ij C m n is kth periodic solution of 39 with φ 0 ij := Φ0 ij Then where l s = T 349 e l C h + s provided s is sufficiently small for some C>0 that is independent of h and s Proof The continuous function Φ solves the discrete equations 350 Φ k+1 Φ k = sδ h μ Φ k+1 Φ k + sτ k+1 where τ k+1 is the local truncation error which satisfies 351 τ k+1 ij M1 h + s for all i j andk for some M 1 0 that depends only on T L x andl y Inparticular we have 35 τ k+1 ij Hence C s Φ C 0T ;C 4 Ω + h Φ 3 L 0T ;C 6 Ω + Φ L 0T ;C 8 Ω 353 M 1 Φ C 0T ;C 4 Ω + Φ 3 L 0T ;C 6 Ω + Φ L 0T ;C 8 Ω Subtracting 39 from 350 yields 354 e k+1 e k = sδ h μ Φ k+1 Φ k μ φ k+1 φ k + sτ k+1

16 84 S M WISE C WANG AND J S LOWENGRUB Multiplying by h e k+1 summing over i and j and applying Green s second identity Proposition eq 1 we have e k+1 e k + e k+1 e k =h s μ Φ k+1 Φ k μ φ k+1 φ k Δh e k+1 +h s τ k+1 e k+1 Φ =h k+1 s 3 φ k+1 3 Δh e k+1 +h s1 ɛ e k+1 Δ h e k+1 +4h s Δ h e k Δh e k+1 +h s Δ he k+1 Δh e k h s τ k+1 e k+1 Dropping the nonnegative term e k+1 e k and using summation-by-parts we have e k+1 e k Φ h k+1 s 3 φ k+1 3 Δh e k+1 s1 ɛ h e k+1 +4h s Δ h e k Δh e k s h Δ h e k+1 +h s τ k+1 e k+1 Using Cauchy s inequality and the pointwise boundedness of both φ k+1 and Φ k+1 we have the following estimate: Φ h k+1 s 3 φ k+1 3 Δh e k+1 s Φ k+1 3 φ k s Δh e k sc 1 e k+1 + s Δh e k+1 where C 1 is independent of s and h Two more applications of Cauchy s inequality yield 358 4h s Δ h e k Δh e k+1 s Δh e k +s Δh e k+1 and 359 h s τ k+1 e k+1 sm h + s + s e k+1 where M := M 1 L xl y Putting these together and dropping the nonpositive term s1 ɛ h e k+1 yields e k+1 e k s C 1 +1 e k+1 +s Δh e k +3s Δh e k s h Δ h e k+1 + sm h + s

17 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 85 Summing over k and using e 0 = 0 weobtain e l l 1 = { e k+1 } e k k=0 l 1 { s C 1 +1 e k+1 +s Δh e k +3s Δh e k+1 k=0 s h Δ h e k+1 + sm h + s } 361 l = s C 1 +1 l e k +5s Δh e k s Δh e l s l h Δh e k + h + s M T s C 1 +1 s l l e k +5s Δh e k l h Δh e k + h + s M T whereweassumel s = T WenowuseLemma310withe k to obtain e l s C l l e k 10α 3α + s 3 h Δh e k 36 + h + s M T where α>0 is arbitrary Choosing α =3/10 yields e l s C l e k l 7 s h Δh e k h + s M T Dropping the nonpositive term s l yields 364 Manipulating we have e l sc h Δh e k and setting C := C /7 l e k + h + s M T 365 e l sc 1 sc l 1 e k + TM 1 sc h + s Provided s<1/c the discrete Gronwall inequality guarantees that 366 e l TM 1+ sc l 1 h + s 1 sc 1 sc

18 86 S M WISE C WANG AND J S LOWENGRUB The coefficient 367 TM 1 sc 1+ sc l 1 1 sc may be bounded by a positive constant that is dependent on T exponentially but otherwise independent of h and s This proves the theorem Remark 31 Energy estimates to the semidiscrete scheme 115 at the continuous-space level can be applied to establish the well-posedness of the PFC model The limit of the numerical solution as time step s 0 is a solution of the PDE system The nonincreasing property of the physical energy 117 indicates a bound in the L 0T; H Ω norm Moreover by testing the numerical scheme with Δφ k+1 we can derive an L 0T; H 4 Ω bound of the numerical solution independent of s with the help of the L estimate of φ k+1 This in turn gives a weak solution of the PFC model by taking the limit of s 0 with the regularity φ L 0T; H Ω L 0T; H 4 Ω Similar energy estimate techniques can be applied to analyze the higher-order derivatives of the numerical solution By testing the numerical scheme with Δ m φ k+1 m 3 we obtain a higher-order regularity of the numerical solution with a uniform s-independent bound for φ L 0T; H m Ω L 0T; H m+3 Ω The analysis of the nonlinear term can be handled in the same way as in the derivation of the weak solution As a result a unique global in time strong solution with m =3canbe proven for the PFC model since passing to the limit as s 0doesnotcauseany difficulty A unique global smooth solution can also be derived for any smooth initial data by taking a higher value of m 4 Conclusions In this paper we have developed and proven convergence of an unconditionally energy-stable finite-difference scheme for the sixth-order phase field crystal PFC equation The method is based on a convex splitting of a discrete energy and is semi-implicit The equation at the implicit time level is nonlinear but is uniquely solvable for any time step The algorithm is first-order accurate in time and second-order accurate in space as is demonstrated in an error estimate Our scheme is based on a generalization of the convex-splitting methodology proposed by Eyre [8] for solving bistable gradient-flow equations Herein we put the framework on a sound complete theoretical footing which was missing in [8] In particular Theorem 35 which is the centerpiece of our work facilitates the energy stability of both conserved and nonconserved schemes in a general way Indeed the theory herein applies to a very broad class of bistable gradient flows and as we show in [19] to certain equations of nongradient-flow type as well In a companion paper [11] we develop an efficient nonlinear multigrid method to solve the unconditionally energy-stable algorithm presented here In fact we demonstrate therein that the nonlinearity of the schemes is not a difficulty We compare solutions of our first-order scheme with those of a new fully second-order accurate multistep numerical scheme also given in [11] In this new scheme the discrete energy is bounded by its initial value for any time step but may not be strictly decreasing from one time step to the next This weak energy stability is still sufficient to guarantee pointwise stability of the scheme and the convergence analysis given here can be modified to apply to this new scheme It remains an open question however whether one can obtain a fully second-order accurate numerical scheme that is both i unconditionally strongly energy stable ie the discrete energy is nonincreasing from one time step to the next and ii unconditionally solvable

19 A SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION 87 Herein we have utilized periodic boundary conditions This usage is for simplicity but it is in no ways a fundamental constraint All of our results hold though in perhaps slightly modified forms assuming homogeneous Neumann boundary conditions and also boundary conditions of mixed periodic-homogeneous Neumann type Indeed in [11] we perform a computation with the latter type of boundary conditions There are a number of other interesting problems we plan to address in the future These include the development of adaptive time step algorithms Such algorithms can be very effective in accelerating PFC simulations since the dynamics typically involves a slowly evolving microstructure punctuated by a few rapid events An accelerated algorithm would make it possible to increase the range of scales capable of being simulated by the PFC model The role of adaptive spatial mesh refinement is less clear due to the periodic oscillations of the phase field variable that characterize the spatial extent of the crystal phase Finally we mention that the convex-splitting framework can be extended to deal with an important generalization of the PFC equation In forthcoming papers [10 19] we analyze and test numerically convex-splitting schemes for the modified phase field crystal MPFC equation which was proposed in [16] The MPFC equation is designed to account for elastic interactions and includes a second-order time derivative which adds a quasiphononic time scale on top of the diffusive time scale in the PFC equation [15 16] Specifically the MPFC equation is 41 β tt φ + t φ = Mφ μ which is a generalized damped wave equation In the MPFC approach atomic positions relax at early times consistent with elasticity theory and the evolution on long time scales is dissipative Because of the presence of elastic waves the MPFC equation can have stricter time step requirements than the PFC and thus the design of efficient algorithms for the MPFCisveryimportant Acknowledgment The authors thank A Voigt and Z Hu for valuable discussions REFERENCES [1] S M Allen and J W Cahn A microscopic theory for antiphase boundary motion and its application to antiphase domain coursening Acta Metall pp [] RBackofenARätz and A Voigt Nucleation and growth by a phase field crystal PFC model Phil Mag Lett pp [3] J Cahn On spinodal decomposition Acta Metall pp [4] J Cahn and J Hilliard Free energy of a nonuniform system I Interfacial free energy J Chem Phys pp [5] M Cheng and J Warren An efficient algorithm for solving the phase field crystal model J Comput Phys pp [6] B Dacorogna Introduction to the Calculus of Variations Imperial College Press London 004 [7] K Elder M Katakowski M Haataja and M Grant Modeling elasticity in crystal growth Phys Rev Lett article [8] D Eyre Unconditionally gradient stable time marching the Cahn-Hilliard equation incomputational and Mathematical Models of Microstructural Evolution J W Bullard R Kalia M Stoneham and L Q Chen eds Mater Res Soc Symp Proc 59 Materials Research Society Warrendale PA 1998 pp [9] D Furihata A stable and conservative finite difference scheme for the Cahn-Hilliard equation Numer Math pp [10] Z Hu S Wise C Wang and J Lowengrub Stable and efficient finite-difference nonlinearmultigrid schemes for the modified phase field crystal equation in preparation

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