IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE * 1. Introduction

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1 Journal of Compuaional Mahemaics Vol.34, No.5, 206, hp:// doi:0.4208/jcm.603-m IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE * Tao Tang Deparmen of Mahemaics, Souhern Universiy of Science and Technology, Shenzhen, Guangdong 58055, China ang@susc.edu.cn Jiang Yang Deparmen of Applied Mahemaics, Columbia Universiy, New York, NY 0027, USA jyanghkbu@gmail.com Absrac I is known ha he Allen-Chan equaions saisfy he maximum principle. Is his rue for numerical schemes? To he bes of our knowledge, he sae-of-ar sabiliy framework is he nonlinear energy sabiliy which has been sudied exensively for he phase field ype equaions. In his work, we will show ha a sronger sabiliy under he infiniy norm can be esablished for he implici-explici discreizaion in ime and cenral finie difference in space. In oher words, his commonly used numerical mehod for he Allen-Cahn equaion preserves he maximum principle. Mahemaics subjec classificaion: 65GXX, 65MXX. Key words: Allen-Cahn Equaions, Implici-explici scheme, Maximum principle, Nonlinear energy sabiliy.. Inroducion This paper is concerned wih he numerical approximaion of he Allen-Cahn equaion wih he iniial condiion u = ϵ2 u f(u), x Ω, (0, T ], (.) u(x, 0) = u 0 (x), x Ω, (.2) and subjecs o he periodic or homogeneous Neumann/Dirichle boundary condiions, where Ω is a bounded domain in R d (d =, 2, 3), u represens he concenraion of one of he wo meallic componens of he alloy, and he parameer ϵ > 0 represens he iner-facial widh. Wihou lose of generaliy, we consider he commonly used double well poenial which gives f(u) = u 3 u. (.3) Roughly speaking, he Allen-Cahn equaion (.) describes regions wih u and u ha grow and decay a he expense of one anoher []. Define he energy funcion in L 2 - space ( ) E(u) = 2 ϵ2 u 2 + F (u) dx, (.4) Ω * Received Augus 8, 204 / Revised version received December 23, 205 / Acceped March 2, 206 / Published online Sepember 4, 206 /

2 452 T. TANG AND J. YANG where F (u) = 4 (u2 ) 2. One of he inrinsic properies of he Allen-Cahn equaion is ha he energy funcion is decreasing wih ime: d E(u) 0, > 0. (.5) d The Allen-Cahn equaion was originally inroduced by Allen and Cahn in [] o describe he moion of ani-phase boundaries in crysalline solids. As he exac soluions of hese phasefield models can no be found, numerical mehods have played an imporan role in various simulaions. In paricular, here has been exensive numerical sudy for approximaing various phase field models, see, e.g., he survey aricles of [9, 0]. One of he imporan numerical aspecs is abou he discree sabiliy of he numerical schemes. For he Allen-Cahn equaion, some recen sabiliy analysis can be found in [4,6,2,4,5]. To he bes of our knowledge, he exising sabiliy analysis for he phase field models has been resriced o he energy seing, see, e.g., [2, 5, 7, 8,, 3], and here have no rigorous l -sabiliy analysis for he numerical mehods. I is known ha he soluions of he Allen-Cahn equaion (.) saisfies he maximum principle, see, e.g., [3]. The primary goal of his paper is o esablish a discree L -sabiliy analogue. More precisely, we will show ha for he implici-explici discreizaion in ime and cenral finie difference in space, he numerical soluions for (.)-(.3) can be bounded by under he condiion ha he iniial daa is bounded by. In oher words, his commonly used numerical mehod for he Allen-Cahn equaion preserves he maximum principle. To demonsrae he main idea, we only consider a regular soluion domain in R d (d =, 2, 3). Wihou lose generaliy, we only consider a uni square in 2D and a cube in 3D. We also use he cenral finie difference o approximae he spaial derivaives and denoe D h as he discree marix of he Laplace operaor. I is known ha he discree marix of he Laplace operaor subjeced wih homogeneous Dirichle boundary condiions in D is given by D h = Λ h := h N N, (.6) where h is he widh of an D uniform mesh. By using he noaion of he Kronecker ensor produc, we can obain he discree marix in 2D: D h = I Λ h + Λ h I, (.7) where I is he N N ideniy marix. Similarly, he discree marix of 3D case can be represened as D h = I I Λ h + I Λ h I + Λ h I I. Independen of he dimension, i can be verified ha he discree marix D h saisfies he following properies: D h is symmeric; D h is negaive semidefinie, i.e., U T D h U 0, U R N ; (.8)

3 Implici-Explici Scheme Preserves he Maximum Principle 453 Elemens of D h saisfy b ii = b < 0, b max i b ij, i N. (.9) j i 2. The Discree Maximum Principle and Energy Sabiliy We firs prove he following useful lemma. Lemma 2.. Le B R N N and A = ai B, where a > 0. If B = (b ij ) saisfies (.9), hen Proof. We firs wrie A in he following equivalen form: A a. (2.) A = (a + b)(i sc), (2.2) where b is given by (.9), s = b/(a + b) <, and marix C = (c ij ) saisfies c ii = 0, max c ij = max b ij i i b, i N. (2.3) j i j i By Gershgorin s circle heorem, i can be verified ha C, ρ(sc) = sρ(c) s <, (2.4) where ρ(c) sands for he specral radius of he marix C. As he inverse of I sc can be represened by he power series of sc, we have A = a + b p=0(sc) p s p C p a + b p=0 a + b s = a, (2.5) where in he las sep we have used he fac ha 0 < s <. This complees he proof of he lemma. The mos convenional approach for solving (.) is o use he sandard implici-explici scheme in ime and cenral finie difference in space: U n+ U n + ((U n ).3 U n ) = ϵ 2 D h U n+, (2.6) τ where τ denoes he ime sep-size, U n represens he vecor of numerical soluion a he = n level, and (U n ).3 = ((U n ) 3, (U n 2 ) 3,, (U n N )3 ) T. 2.. The maximum principle Theorem 2.. Consider he Allen-Cahn problem (.)-(.3) wih periodic or homogeneous Neumann/Dirichle boundary condiions. If he iniial value is bounded by, i.e., max x Ω u 0 (x), hen he numerical soluion of he fully discree scheme (2.6) is also bounded by in he sense ha U n for all n > 0, provided ha he ime sepsize saisfies 0 < τ 2.

4 454 T. TANG AND J. YANG Proof. We prove our claim by inducion. Obviously, U 0 u 0. We assume U m and will verify he resul is rue for U m+. I follows from he scheme (2.6) ha U m+ = (I τϵ 2 D h ) (U m + τ(u m (U m ).3 )). (2.7) By Lemma 2., we have (I τϵ 2 D h ). (2.8) Noe ha each elemen of U m + τ(u m (U m ).3 ) is of he form g(x) = x + τ(x x 3 ). I can be verified ha if 0 < τ 2 hen g (x) 0 for x [, ]. This gives ha max g(x) = g() = ; min x g(x) = g( ) =, x which implies ha g =. Consequenly, we can conclude ha U m + τ(u m (U m ).3 ) if U m. (2.9) This, ogeher wih (2.7) and (2.8), gives U m+ (I τϵ 2 D h ) U m + τ(u m (U m ).3 ). (2.0) This complees he proof The discree energy sabiliy Subjeced wih he periodic or homogeneous Neumann/Dirichle boundary condiions, we have ( E(u) = 2 ϵ2 u 2 + F (u)) dx = Ω Ω ( ) 2 ϵ2 u u + F (u) dx, (2.) where E(u) is defined by (.4). The discree energy funcion can be represened by he discree Laplace operaor D h given below where d is he number of dimension. E h (U) = h d( ϵ2 2 U T D h U + N i= 4 (U 2 i ) 2), (2.2) Theorem 2.2. Consider he Allen-Cahn problem (.)-(.3) wih periodic or homogeneous Neumann/Dirichle boundary condiions. If he iniial value is bounded by, i.e., max x Ω u 0 (x), hen he numerical soluions obained by he scheme (2.6) saisfies he discree energy decreasing propery: provided ha he ime sep-size saisfies 0 < τ 2. E h (U n+ ) E h (U n ), (2.3)

5 Implici-Explici Scheme Preserves he Maximum Principle 455 Proof. Taking he difference of he discree energy beween wo consecuive ime level gives E h (U n+ ) E h (U n ) = hd 4 N i= [ ((U n+ i ) 2 ) 2 ((U n i ) 2 ) 2] ϵ2 h d ( (U n+ ) T D h U n+ (U n ) T D h U n). (2.4) 2 Noe ha for all a, b [, ]: (b 3 b)(a b) + (a b) 2 4 [(a2 ) 2 (b 2 ) 2 ]. (2.5) I follows from Theorem 2. ha U n+, U n wih 0 < τ 2. This fac, ogeher wih (2.4), gives h d E h (U n+ ) E h (U n ) N i= [ ((U n i ) 3 U n i )(U n+ i U n i ) + (U n+ i U n i ) 2] ϵ2 h d ( (U n+ ) T D h U n+ (U n ) T D h U n). (2.6) 2 Taking L 2 inner produc for (2.6) wih (U n+ U n ) T yields N i= [ ((Ui n ) 3 Ui n )(U n+ i Ui n ) + n+ (Ui Ui n ) 2] τ = ϵ 2 (U n+ U n ) T D h U n+. (2.7) Since he discree Laplace operaor D h is symmeric, we can rewrie he righ-hand side of (2.7) as = ϵ2 2 ϵ 2 ((U n+ U n ) T D h U n+ ( (U n+ ) T D h U n+ (U n ) T D h U n) ( ) + ϵ2 (U n+ U n ) T D h (U n+ U n ). (2.8) 2 Consequenly, combining (2.6)-(2.8) gives E h (U n+ ) E h (U n ) ϵ2 h d 2 (U n+ U n ) T D h (U n+ U n ) 0. (2.9) Since D h is negaive semidefinie, (2.3) follows immediaely from he above inequaliy. 3. Uncondiionally Sabilized Implici-Explici Scheme I is shown in he previous secion ha he commonly used scheme (2.6) is condiionally sable. In he following numerical secion we will show ha he sabiliy condiion 0 < τ /2 is boh necessary and sufficien. To obain an uncondiionally sable implici-explici scheme,

6 456 T. TANG AND J. YANG we can add an exra perurbaion erm which is consisen wih he runcaion error. example, we can follow [2] o give a modified scheme: For where β > 0 is a consan. U n+ U n + ((U n ).3 U n ) + β(u n+ U n ) = ϵ 2 D h U n+, (3.) τ Theorem 3.. Consider he Allen-Cahn problem (.)-(.3) wih periodic or homogeneous Neumann/Dirichle boundary condiions. If he iniial value is bounded by, i.e., max x Ω u 0 (x), hen he numerical soluions obained by he scheme (3.) saisfy U n, E h (U n+ ) E h (U n ), provided ha β + τ 2, (3.2) where he discree energy E h is defined by (2.2). In paricular when β 2, he numerical scheme (3.) is uncondiionally poinwise sable and energy sable. Proof. The proof is similar o ha of Theorems 2. and 2.2, and will be omied here. 4. Error Analysis In his secion, we will esablish error esimaes for he fully discree formulaion of he sandard implici-explici scheme (2.6). The error U n is defined as E n = U( n ) U n, (4.) where U( n ) is he exac soluion of (.)-(.3). Firs, we define he local runcaion error T n for he scheme (2.6) by T n = (I τϵ 2 D h )U( n ) (U( n ) + τ(u( n ) (U( n )).3 )). (4.2) Taking one-dimension for insance, i.e. Ω = (a, b), i is rivial o compue Tj n by Taylor expansion as ( ( = τ τ u 2 ( n, x j ) u( n, x j )u( n, x j ) u 2 ( n, x j )) u( n + τθ n, x j ) T n j + τ 2 ) 2 u( n + τθ2 n, x j ) ϵ 2 h2 2 x 4 u( n, x j + hξ j ), (4.3) where u( n, x j ) is he exac soluion a ( n, x j ) and he coefficiens θ n, θ n [0, ] and ξ j [, ]. If he exac soluion u is sufficien smooh, a leas u C 2 (0, T ; C 4 [a, b]), hen he scheme (2.6) is consisen. Hence, i follows ha T n C(T, ϵ)τη for n T, (4.4) where η = τ + h 2 and C = C(T, ϵ, u) is a posiive consan depending on T, ϵ and he exac soluion u bu no depending on τ or h. Based on he smoohness and consisency assumpion, we hen have

7 Implici-Explici Scheme Preserves he Maximum Principle 457 Theorem 4.. Consider he Allen-Cahn problem (.)-(.3) wih periodic or homogeneous Neumann/Dirichle boundary condiions. Assume ha soluion u(x, ) is smooh and he scheme (2.6) is consisen in he sense of (4.4). If he iniial value is smooh and bounded by, i.e., max x Ω u 0 (x), we have E n e κ n (τ + h 2 ) for n T, (4.5) provided ha τ 0.5, where κ = max{c(t, ϵ), 2}, and C(T, ϵ) is given by (4.4). Proof. We rewrie he scheme (2.6) as Subracing (4.6) from (4.2) obains (I τϵ 2 D h )U n = U n + τ(u n (U n ).3 ). (4.6) (I τϵ 2 D h )E n = E n + τ[(u( n ) (U( n )).3 ) (U n (U n ).3 )] + T n. (4.7) From he fundamenal inequaliy i is easy o derive (a a 3 ) (b b 3 ) 2 a b for a, b [, ], (4.8) (U( n ) (U( n )).3 ) (U n (U n ).3 ) 2 E n, (4.9) due o boh U( j ) and U j lying in [, ] for j n. Combining (4.2) and (4.9) and using Lemma 2., we have from (4.7) where κ = max{c, 2}. By inducion i follows E n ( + κτ) E n + κτη, (4.0) n E n ( + κτ) n E 0 + κτη ( + κτ) j e κn ( E 0 + η), (4.) j=0 which is he desired resul (4.5) as E 0 = Numerical Tess In his secion, we presen some numerical experimens o verify he heoreical resuls obained in he previous secions, paricularly on he energy sabiliy and numerical maximum principle. Since our analysis is independen of dimensions, for simpliciy we only consider onedimensional problems for (.) wih homogeneous Neumann boundary condiion. The iniial condiion is chosen as u 0 (x) = rand( ) , where rand( ) represens a random number on each poin in [0, ]. The parameer ϵ 2 is 0.00, he compuaion domain is [0, ] and he mesh size in space is h = 0.0. We firs consider he sandard implici-explici scheme (2.6). Fig 5. plos he energy curves for several values of τ, and i is found ha he energy blows up quickly when τ = 3.

8 458 T. TANG AND J. YANG τ= τ= τ= 0 x 06 8 τ= Fig. 5.. s for scheme (2.6) wih differen ime seps τ = 0.5, 0.75,, 3. Energy β= τ=0.5 τ= τ=3 5 τ= τ=.05 τ= Fig Energy curves and maximum values for (3.) wih β = and ime seps τ = 0.5,, 3. Fig. 5. plos he maximum soluion values agains ime, and he numerical resuls are in excellen agreemen of our heoreical analysis. More precisely, he maximum principle is preserved

9 Implici-Explici Scheme Preserves he Maximum Principle 459 Energy β=2 τ= τ= τ= τ= τ= τ= Fig Energy curves and maximum values for (3.) wih β = 2 and ime seps τ = 0.5,, 3. for τ = 0.5 and is violaed when τ = 0.75,, 3. Fig. 5.2 gives he numerical resuls obained using he modified scheme (3.) wih β =. Several ime sep-sizes τ are used. I is seen when he requiremen β + /τ 2 is no saisfied wih β = and τ = 3 he maximum principle is violaed. Finally, we change β from o 2 and i is observed from Fig. 5.3 ha he corresponding scheme becomes uncondiionally sable. This is in good agreemen wih he resuls of Theorem Concluding Remarks This works provides a heoreical framework for analyzing he l -sabiliy for he approximae soluions o he Allen-Cahn equaions. Alhough similar heoreical resuls do no hold for phase field models which involve biharmonic operaors, we are considering some weak version of he l -sabiliy. Noe ha for simpliciy, we only consider a special case of poenial funcional (.3). For general cases, he free energy F (u) has a double well form wih minimum a γ and γ saisfying Moreover, F (u) (i.e. f(u)) saisfies he following condiions F (γ) = F ( γ) = 0. (6.) F (u) < 0, u (, γ), F (u) > 0, u (γ, ). (6.2) In his case, he implici-explici scheme will become U n+ U n + f(u n ) = ϵ 2 D h U n+. (6.3) τ

10 460 T. TANG AND J. YANG The heoreical resuls in previous secions hold wih some minor changes. For example, Theorem 2. will become Theorem 6.. Consider he Allen-Cahn problem (.)-(.3) wih periodic or homogeneous Neumann/Dirichle boundary condiions. If he iniial value is bounded by γ, i.e., max x Ω u 0 (x) γ, hen he fully discree scheme (6.3) is also bounded by γ in he sense ha U n γ for all n > 0, provided ha he ime sep-size saisfies τ max F (x). (6.4) x [ γ,γ] To prove above heorem, he main difference is ha funcion g(x) should be changed o g(x) = x τf (x). Oher pars of he proof are basically he same. Finally, i is poined ou ha he numerical scheme considered in his work are all firs order in ime, which is verified by (4.5). As a resul, we can say ha he maximum principle holds for a class of firs order schemes. I remains open o see wheher he maximum principle is sill rue for high order numerical schemes. This will be an ineresing bu difficul issue. Acknowledgmens. The auhors hank Dr Xinlong Feng for many useful discussions and for providing assisance in numerical simulaions. References [] S.M. Allen and J.W. Cahn, A microscopic heory for aniphase boundary moion and is applicaion o aniphase domain coarsening, Aca Meall, 27 (979), [2] D.J. Eyre, An uncondiionally sable one-sep scheme for gradien sysems, June 998, unpublished, hp:// [3] L.C. Evans, H.M. Soner and P.E. Souganidis, Phase ransiions and generalized moion by mean curvaure, Commun. Pure Appl. Mah., 45 (992), [4] X. Feng and A. Prohl, Numerical analysis of he Allen-Cahn equaion and approximaion for mean curvaure flows, Numer. Mah., 94() (2003), [5] X. Feng, H. Song, T. Tang and J. Yang, Nonlinearly sable implici-explici mehods for he Allen-Cahn equaion, Inverse Problems and Image, 7 (203), [6] X. Feng, T. Tang and J. Yang, Sabilized Crank-Nicolson/Adams-Bashforh schemes for phase field models, Eas Asian Journal on Applied Mahemaics, 3 (203), [7] X. Feng, T. Tang and J. Yang, Long ime numerical simulaions for phase-field problems using p-adapive specral deferred correcion mehods, SIAM J. Sci. Compu. 37 (205), A27-A294. [8] H. Gomez and T. Hughes, Provably uncondiionally sable, second-order ime-accurae, mixed variaional mehods for phase-field models, J. Compu. Phys., 230 (20), [9] L. Golubovic, A. Levandovsky and D. Moldovan, Inerface dynamics and far-from-equilibrium phase ransiions in mulilayer epiaxial growh and erosion on crysal surfaces: Coninuum heory insighs, Eas Asian J. Appl. Mah., (20), [0] J. Kim, Phase-field models for muli-componen fluid flows, Commun. Compu. Phys., 2 (202), [] Z. Qiao, Z. Zhang and T. Tang, An adapive ime-sepping sraegy for he molecular beam epiaxy models, SIAM J. Sci. Compu., 33 (20), [2] J. Shen and X. Yang, Numerical approximaions of Allen-Cahn and Cahn-Hilliard equaions, Discre. Conin. Dyn. Sys., 28 (200), [3] C. Xu and T. Tang, Sabiliy analysis of large ime-sepping mehods for epiaxial growh models, SIAM J. Numer. Anal., 44 (2006),

11 Implici-Explici Scheme Preserves he Maximum Principle 46 [4] X. Yang, Error analysis of sabilized semi-implici mehod of Allen-Cahn equaion, Discree Conin. Dyn. Sys. Ser. B, :4 (2009), [5] J. Zhang and Q. Du, Numerical sudies of discree approximaions o he Allen-Cahn equaion in he sharp inerface limi, SIAM J. Sci. Compu., 3:4 (2009),