Convergence analysis for second-order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations

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1 Received: 4 December 06 DOI: 0.00/mma.4497 RESEARCH ARTICLE Convergence analysis for second-order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations Zhen Guan John Lowengrub Cheng Wang Department of Mathematics, University of California, Irvine, 9697, CA, USA Department of Mathematics, University of Massachusetts, Dartmouth, 0747, MA, USA Correspondence Cheng Wang, Department of Mathematics, University of Massachusetts, Dartmouth, MA 0747, USA. cwang@umassd.edu Communicated by: D. Zeidan Funding information NSF-CHE, Grant/Award Number: 0358 ; NSF-DMR, Grant/Award Number: 05409; NSF-DMS, Grant/Award Number: 773; NSF-DMS, Grant/Award Number: FRG ; NSF-DMS, Grant/Award Number: ; NSFC, Grant/Award Number: 78 MOS Classification: 65M; 65R0 In this paper, we provide a detailed convergence analysis for fully discrete second-order in both time and space numerical schemes for nonlocal Allen-Cahn and nonlocal Cahn-Hilliard equations. The unconditional unique solvability and energy stability ensures l 4 stability. The convergence analysis for the nonlocal Allen-Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn-Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H inner-product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori W, bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang eg, 964, instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an Os 3 + h 4 convergence in l 0, T; l norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s Ch. Here, we also prove convergence of the scheme in the maximum norm under the same constraint. KEYWORDS convergence analysis, energy stability, higher order asymptotic expansion, nonlocal Allen-Cahn equation, nonlocal Cahn-Hilliard equation, second-order numerical scheme INTRODUCTION In this paper, our primary goal is to develop a detailed convergence analysis of fully discrete second-order in both time and space numerical schemes for nonlocal Allen-Cahn nac and nonlocal Cahn-Hilliard nch equations, which are integro-partial-differential equations IPDEs defined as follows: t φ = Mφw nac,. and where t φ = Mφ w nch,. w = δ φ E = φ 3 + γ c φ γ e φ +J φ J φ Copyright 07 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/mma Math Meth Appl Sci. 07;40:

2 GUAN ET AL and Mφ 0 is the mobility. These equations are defined in 0, T] Ω, with the initial condition φ0, x =φ 0 x,.4 where Ω is a rectangular domain in R n. The solution φx, t is Ω-periodic in space. The convolution kernel J is also Ω-periodic and even. By J φ, we mean the circular, or periodic, convolution defined as J φ x = Ω Jy φx y dy = Ω Jx y φy dy,.5 in which γ c 0andγ e 0 are system parameters. For the convenience of analysis, we use the form γ c γ e instead of merging them together. The term w defined in.3 is the chemical potential variational derivative relative to the energy { Eφ = Ω 4 φ4 + γ c γ e φ + } Jx yφx φy dy dx..6 4 Ω Equations. and. are L nac and H nch flows of Eφ and their periodic solutions dissipate E at the rate d t Eφ = M w, d L t Eφ = M w, respectively. L Equations. and. are members of a more general family of IPDE models: t φ = Mφw general nac,.7 and t φ = Mφ w general nch,.8 where w general = δ φ E = F φ+j φ J φ..9 The energy E general φ associated with this model is defined as E general φ = {Fφ+ } Jx yφx φy dy dx,.0 Ω 4 Ω where Fφ is the local energy density and it is usually nonlinear. The classical Allen-Cahn AC and Cahn-Hilliard CH equations are the approximation to this type of IPDE models, whose free energy -3 is E local φ = Gφ+ ε Ω φ dx,. and the corresponding dynamical equations are t φ = Mφδ φ E local = Mφ G φ ε Δφ AC,. and t φ = Mφ δ φ E local = Mφ G φ ε Δφ CH..3 The approximation is established as follows. 4-8 By Taylor's expansion, J φ J 0 φ + J Δφ. Here, J 0 = Ω Jxdx and J = Ω Jx x dx J is the second moment of J. Under the assumption of periodic boundary conditions, we have the following approximation: φ, J φ L φ, J 0 φ + J Δφ = J 0 φ, + J φ dx..4 L L 4 Ω On the other hand, the energy E general is equivalent to E general φ =Fφ, L φ, J φ L..5 Denote Gφ =Fφ J 0 φ and ε = J, we can obtain E local E general. General nch and nac equations such as Equations.7 and.8 have been widely used in many fields ranging from physics and material science to biology, finance, and image processing. In material science, Equation.8 and other closely related models arise as mesoscopic models of interacting particle systems. 7,9,0 Equation.7 is also used to model phase transition. For instance, in dynamic density functional theory model, 4,9 the interaction kernel J = c x, y φ ref is the -particle direct correlation function, φ represents the mesoscopic particle density, and φ ref is the average density. Another example is the

3 6838 GUAN ET AL. structural phase field crystal model. It is developed to model the evolution of materials in diffusive scale, while maintaining atomic resolution. The structural phase field crystal model and its extensions have been used in many researches for material science since. 3-5 Readers are referred to previous studies 6,8,6-0 for further details. In biology, Equation.8 has been used to model the tissue growth that covers both bulk and cellular levels. In these models, J has been used to model interactions among cells and extracellular matrix. -4 Equation.7 has also been used in mathematical models of finance. In this case, J arises from an expectation taken with respect to a particular measure that is used in the model for option pricing. 5 In the modeling for image segmentation with nch equation, J is interpreted as the attracting force. 6,7 Readers are referred to previous works 7-35 for theoretical studies of general nch equations and another studies 5,,36 for general nac equations. The nch equation has also been extended to model hydrodynamics with the coupling of Navier-Stokes type dynamics There are a few works dedicated to numerical simulation of, or numerical methods for, equations like. and.. Anitescu et al 40 considered an implicit-explicit time stepping framework for a nonlocal system modeling turbulence, where, as here, the nonlocal term is treated explicitly. References Du et al and Zhou et al 4,4 address finite element approximations in space of nonlocal peridynamic equations with various boundary conditions. In Bates et al, 43 a finite-difference method for Equation. with nonperiodic boundary conditions is applied and analyzed. Reference Hartley et al 44 uses a spectral-galerkin method to solve a nac-type problem, like Equation., but with a stochastic noise term and equation modeling heat flow. The spectral method is also studied for nac equation. 45 For other references for equations like., see previous works. 7,6,46,47 Afirst-order convex splitting for Equation. scheme was introduced and analyzed in Guan et al. 48 The scheme is also extended to nonlocal hydrodynamics. 49 A second-order convex splitting scheme for the general system Equation.8 was introduced in Guan et al 50 ; its unconditional energy stability and unique solvability were presented. Here, we present a detailed convergence analysis of fully discrete second-order convex splitting schemes using the specific forms of Equations. and.. We prove convergence in both the l and l spatial norms. We note that this convergence analysis is much more challenging than that of the first-order scheme, primarily because of its complicated form for the nonlinear term and lack of higher order diffusion term. The present work focuses on the D case but can be straightforwardly extended to 3D. Although the result cannot be easily extended to more general nonlinear local densities F, our results are still useful because polynomial local density functions are widely used. The outline of the paper is given as follows. In Section, we define the fully discrete second-order scheme and give some of its basic properties, including energy stability and unique solvability. The second-order convergence analysis for the nac equation is presented in Section 3. In Section 4, we provide a higher order consistency analysis for the nch equation, up to order Os 3 + h 4. In Section 5, we give the details of the convergence analyses for the nch equation, in both l 0, T; l and l 0, T; l norms. Finally, some numerical results are presented in Section 6, which confirm convergence of the schemes. Some concluding remarks are given in Section 7. THE SECOND-ORDER CONVEX SPLITTING SCHEMES. The discrete periodic convolution and useful inequalities Denote Ω to be a rectangular domain in R. Assume convolution kernel J satisfies A. J = J c J e,wherej c, J e are smooth, Ω-periodic, and nonnegative. A. J c and J e are even, ie, J α x, x =J α x, x,forallx, x R, α = c, e. A is used in the convexity analysis for the energy, and A is the result of periodic convolution. The energy.6 is equivalent to Eφ = 4 φ4 + γ c γ e + J L 4 Equations. and. can also be rewritten as φ L φ, J φ L.. t φ = Mφaφφ J φ, nac,. where t φ = aφmφ φ Mφ J φ, nch,.3 aφ =3φ + γ c γ e + J..4

4 GUAN ET AL We refer to aφ as the diffusive mobility or just the diffusivity. To make Equation. positive diffusive and nondegenerate, we require 30 γ c γ e + J = γ 0 > 0,.5 in which aφ > 0. We will assume that.5 holds in the sequel. In this paper, we focus on Equations. and., which are a special case of general nac and nch equations. They can be obtained by denoting local energy density F as Fφ = 4 φ4 + γ c γ e φ,.6 where γ c, γ e 0 are constants. By rewriting the nonlocal part of E general as Ω Ω Jx yφxφydydx = 4 Ω Ω Jx yφx φy dydx Ω J φ xdy,.7 and γ c γ e = γ c γ e + J,.8 we can obtain energy.6. This local density is the approximation to a regular solution model with free energy Fφ =θ[φlogφ+ φlog φ] θ c φ φ,.9 where θ and θ c represent the absolute and critical temperatures, respectively.,5-53 System parameters γ c γ e areusedtoadjust the approximation for different reference states of φ. We need spaces: m n is the space of cell-centered grid or lattice functions, and m n is the space of vertex-centered grid functions. The precise definitions can be found in Appendix A. The spaces and the following notations have straightforward extensions to 3 dimensions. Suppose φ m n is periodic and f m n is periodic. The D grid inner-products, [ ] ew, [ ] ns are defined in Appendix A. The discrete convolution operator [ f φ ] m n m n m n is defined component-wise as [ f φ ]i,j = h m n k= l= f k+,l+ φ i k,j l..0 Note very carefully that the order is important in the definition of the discrete convolution [ ]. The next result follows from the definition of the discrete convolution and simple estimates. The proof is omitted. Lemma.. If φ, ψ m n are periodic and f m n is periodic and even, ie, f i+ = f i+,foralli, j Z,then [ ] [ ] φ f ψ = ψ f φ.. If, in addition, f is nonnegative, then [ ] φ f ψ [ f ] α φ φ + α ψ ψ, α >0.. The following lemma is cited from Guan et al 48 ; the detailed proof can be found there. Lemma.. Suppose φ m n is periodic. Assume that f CperΩ is even and define its grid restriction via f i+ f p i+, p j+,sothatf m n. Then for any α>0,wehave,j+ = h [ f ψ ] Δ h φ C α ψ + α hφ,.3 where C is a positive constant that depends on f but is independent of h.. Semi-discrete convex splitting schemes The second-order in time convex splitting scheme for the nac equation. and nch equation. has been proposed in a recent article, 50 which was proven to be unconditionally solvable and unconditionally energy stable. These schemes follow the convex properties of the energy 48 :

5 6840 GUAN ET AL. Lemma.3. There exists a nonnegative constant C such that Eφ+C 0 for all φ L 4 perω. More specifically, 8 φ4 Eφ+ γ c γ e J e Ω,.4 L 4 φ Eφ+ γ γ e J e Ω..5 L 4 If γ e = 0, theneφ 0 for all φ LperΩ. 4 Furthermore, the energy. can be written as the difference of convex functionals, ie, E = E c E e,where E c φ = 4 φ4 + γ c + J c φ,.6 L 4 L E e φ = γ e + J c + J e φ + L φ, J φ L..7 Lemma.3 shows the estimate of φ in l and l 4, as well as a splitting that separates the energy into convex and concave parts. The decomposition above is not unique, but Equations.6 and.7 will allow us to separate the nonlocal and nonlinear terms, treating the nonlinearity implicitly and the nonlocal term explicitly, without sacrificing numerical stability. To motivate the fully discrete scheme that will follow later, we here present semi-discrete version and briefly describe its properties. For the sake of simplicity, we denote the mobility Mφ to be constant for Equation. and Mφ for Equation.. For the nonconstant mobility case, the second-order algorithm is proposed in Guan et al 50 ; the unique solvability and unconditional energy stability are standard, and the convergence proof can be achieved with similar idea, while with many more technical details. A second-order in time convex splitting scheme for the nac equation. and nch equation. can be constructed as follows: Given φ k C perω,findφ k+, w k+ C perω such that φ k+ φ k = Msw k+, nac equation,.8 φ k+ φ k = sδw k+, nch equation,.9 w k+ = η φ k,φ k+ + J c +γ c φ k+ J c + J e + γ e φ k+ J φ k+,.0 η φ k,φ k+ = φ k +φ k+ φ k + φ k+,. 4 φ k+ = φ k + φ k+, φ k+ = 3 φk φk,. where s > 0 is the time step size. This scheme respects the convex splitting nature of the energy E given in.6 and.7: The contribution to the chemical potential corresponding to the convex energy, E c, is treated implicitly, and the part corresponding to the concave part, E e, is treated explicitly. Eyre 54 is often credited with proposing the convex splitting methodology for the Cahn-Hilliard and Allen-Cahn equations. The idea is, however, quite general and can be applied to any gradient flow of an energy that splits into convex and concave parts. See, for example, previous studies Moreover, the treatment of the nonlinear term, the convex and concave diffusion terms, given by. and., respectively, follows the methodology in an earlier work 59 to derive a second-order accurate convex splitting scheme for the phase field crystal model. Other related works can also be found in Baskaran et al 60 andshenetal. 6 We have the following a priori energy law for the solutions of the second-order scheme.9 to.. The statement for the fully discrete version appears later in Section.4. Their proof can be found in a recent article. 50 Theorem.4. Suppose the energy Eφ is defined in Equation.. For any s > 0, the second-order convex splitting scheme,.8 or.9, with.0 to., has a unique solution φ k+, w k+ C perω. Moreover, by denoting a pseudo energy φ k,φ k+ = E φ k+ + J c +J e + γ c + γ e J φ k+ φ k,φ k+ φ k L, φk+ φ k L.3

6 GUAN ET AL. 684 we have φ k+ φ k, = 0 for any k and φ k,φ k+ + Ms w k+ φ k,φ k, nac equation,.4 L φ k,φ k+ + s w k+ φ k,φ k, nch equation..5 L Also note that the remainder term in the pseudo energy.3 is nonnegative. This implies that the energy is bounded by the initial energy, for any s > 0: E φ k+ φ k,φ k+ φ,φ 0 E φ 0 by taking φ = φ 0. Therefore, we say that the scheme is unconditionally energy stable. In the sequel, we will propose fully discrete versions of these schemes and provide the corresponding analysis..3 Discrete energy and the fully discrete convex splitting schemes We begin by defining a fully discrete energy that is consistent with the energy. in continuous space. In particular, the discrete energy F m n R is given by Fφ = 4 φ4 4 + γ c γ e φ + [J ] φ h φ [J φ],.6 where J = J c J e,andj α m n, α = c, e, are defined via the vertex-centered grid restrictions J α i+,j+ Lemma.5. Suppose that φ m n is periodic and define F c φ = 4 φ4 4 + [J c ] + γ c F e φ = [J c ] + [J e ] + γ e φ + h Then F c and F e are convex, and the gradients of the respective energies are = J α p i+, p j+. φ,.7 φ [J φ]..8 δ φ F c = φ 3 + [J c ] + γ c φ, δ φ F e = [J c ] + [J e ] + γ e φ + [J φ]..9 Hence F, as defined in.6, admits the convex splitting F = F c F e. Proof. F c is clearly convex. To see that F e is convex, we make use of the estimate., and observe that d ds F eφ + sψ s=0 0, for any periodic ψ m n. The details are suppressed for brevity. We now describe the fully discrete schemes in detail. The scheme can be formulated as follows: Given φ k m n periodic, find φ k+, w k+ m n periodic so that φ k+ φ k = Msw k+, nac equation,.30 φ k+ φ k = sδ h w k+, nch equation,.3 w k+ = η φ k,φ k+ + [J c ] + γ c φ k+ [J c ] + [J e ] + γ e φ k+ [ J φ k+ ],.3 in which Δ h is the standard 5-point discrete Laplacian operator, η φ k,φ k+, φ k+/, φ k+ are given by. and., respectively..4 Unconditional solvability and energy stability Now, we show that the convexity splitting is translated into solvability and stability for our scheme, both.30 and.3. The basic method for the proof of the following result was established in Guan et al 48 andguanetal 50 see also other studies and we therefore omit it for brevity.

7 684 GUAN ET AL. First, we present the theorem regarding the unique solvability of scheme.3. Theorem.6. The scheme.3 is discretely mass conservative, ie, φ k+ φ k = 0, forallk 0, and uniquely solvable for any time step size s > 0. Next, we present the discrete version of Lemma.3. Lemma.7. Suppose that φ m n is periodic and the discrete energy F is defined in Equation.6. There exists a nonnegative constant C such that Fφ+C 0. More specifically, 8 φ4 4 Fφ+γ c γ e [J e ] Ω,.33 φ Fφ+γ c γ e [J e ] Ω The following result is a discrete version of Theorem.4; its proof can be found in Guan et al. 50 Theorem.8. Suppose the energy Fφ is defined in Equation.6; assume φ k+,φ k m n are periodic and they are solutions to the scheme.30 for the nac equation or.3 for the nch equation. Then for any s > 0, φ k,φ k+ + Ms w k+ φ k,φ k, nac equation,.35 φ k,φ k+ + s h w k+ φ k,φ k, nch equation,.36 with the discrete pseudo energy φ k,φ k+ given by φ k,φ k+ = F φ k+ + [J c ] + [J e ] + γ c + γ e 4 φk+ φ k + h [ J φ k+ φ k] φ k+ φ k Most importantly, the remainder term in the pseudo energy.37 is nonnegative. This implies that the energy is bounded by the initial energy, for any s > 0:F φ k+ φ k,φ k+ φ,φ 0 F φ 0 by taking φ = φ 0. Putting Lemma.7 and Theorem.8 together, we immediately get the following results. The proof can be found in Guan et al. 48 Corollary.9. Suppose that { φ k, w k} l k= [ m n] are a sequence of periodic solution pairs of the scheme.30 for the nac equation or.3 for the nch equation, with the starting values φ 0. Then, for any k l, 4 8 φk φ 0 + γ c γ e [J e ] Ω, 4.38 φk φ 0 + γ c γ e [J e ] Ω Theorem.0. Suppose Φ CperΩ, r wherer Z + is sufficiently large, and set φ 0 = Φp i,j i, p j. Suppose that { φ k, w k} l k= [ m n] are a sequence of periodic solution pairs of the scheme.30 for the nac equation or.3 for the nch equation, with the starting values φ 0. There exist constants C 3, C 4, C 5 > 0, which are independent of h and s, such that max k l φk 4 C max k l φk C Convergence result in l l and l l norm We conclude this subsection with the statement of local-in-time error estimates for our second-order convex splitting schemes, including both.30 for the nac equation and.3 for the nch equation, in dimensions. The detailed proof is given in the

8 GUAN ET AL next 3 sections. The extension of the proofs to 3 dimensions is omitted for the sake of brevity; see Remark 5.4 for some of the details. For the nac equation, the existence and uniqueness of solution to the IPDE. has been studied by Bates and Fife in previous studies.,6 Using similar techniques, one can establish the existence and uniqueness of a smooth, periodic solution to the IPDE. with smooth periodic initial data. The second-order convergence of the scheme.30 for the nac equation is stated in the following theorem. Theorem.. Given smooth, periodic initial data Φx, x, t = 0, we assume that the unique, smooth, periodic solution for the nac equation. is given by Φx, y, t on Ω for 0 < t T, for some T <. DenoteΦ k = Φ k p i,j i, p j,andset e k i,j = Φk i,j φk i,j,whereφk i,j m n is kth periodic solution of.30 with φ 0 i,j = Φ0. Then, provided s and h are sufficiently i,j small, we have el C h + s,.4 for all positive integers l, such that ls T, where C > 0 is independent of h and s. For the nch equation, the existence and uniqueness of a smooth, periodic solution to the IPDE. with smooth periodic initial data may be established using techniques developed by Bates and Han in previous studies. 9,30 In the following pages, we denote this IPDE solution by Φ. Motivated by the results of Bates and Han, and based on the assumptions in the introduction, the global in time L bound is available for Φ. Moreover, with certain analysis techniques, the higher order H m for any m regularity could also be derived for Φ, with local-in-time existence, and with the assumption of smooth periodic initial data. Therefore, it will be reasonable to assume that Φ L 0,T;L 4 + Φ L 0,T;L + Φ L 0,T;L < C,.43 for any T > 0, and therefore also, using a consistency argument, that max Φk 4 +max Φk +max hφ k < C,.44 k l k l after setting Φ k i,j = Φp i, p j, t k,wherecis independent of h and s and t k = k s. The IPDE solution Φ is also mass conservative, meaning that, for all 0 t T, Ω Φx, 0 Φx, t dx = 0. For our numerical scheme, on choosing φ 0 i,j = Φ0 i,j,wenotethat φ k Φ 0 = 0, for all k 0. But, unfortunately, φ k Φ k 0 in general. On the other hand, by consistency, βt = Φx, tdx h Φt Φ 0 xdx h Φ 0,.45 Ω L L Ω L L for some C > 0 that is independent of k and h. A more detailed consistency analysis shows that βt Ch, t 0, and the estimates for all its higher order derivatives are available. Then we have k l L L h φ k Φ k = β k = β t k, βk Ch,.46 for all k l. Weset Φ, t =Φ, t+βt and observe φ k Φ k = 0andalso max Φ k 4 +max Φ k +max h Φ k < C..47 k l k l Finally, the assumptions on the continuous kernel J, specifically.5, and the consistency of the discrete convolution imply that [J c ] + γ c [J e ] γ e > 0. Furthermore, we make the following assumption to simplify the convergence analysis: k l B c = [J c ] + γ c, B e = [J c ] + [J e ] + γ e, and B c 3B e = α 0 > 0,.48 for some α 0 that is independent of h, provided that h is sufficiently small. However, numerical evidence indicates that our scheme converges at the same rate when α 0 0. Theorem.. Given smooth, periodic initial data Φx, x, t = 0, suppose the unique, smooth, periodic solution for the IPDE. is given by Φx, y, t on Ω for 0 < t T, for some T <. Define Φ k i,j as above and set ek i,j = Φk i,j φk i,j,where φ k i,j m n is kth periodic solution of.3 with φ 0 i,j = Φ0. Then, provided s and h are sufficiently small with the linear i,j refinement path constraint s Ch, with C any fixed constant, we have

9 6844 GUAN ET AL. el C h + s,.49 for all positive integers l, such that ls T, where C > 0 is independent of h and s. Theorem.3. Under the assumptions of Theorem., we also have optimal order convergence of the numerical solution of the scheme.3 in the l norm. Namely, if s and h are sufficiently small, for all positive integers l, such that ls T, we have el C h + s,.50 where C > 0 is independent of h and s. 3 THE SECOND-ORDER CONVERGENCE ANALYSIS FOR THE NAC EQUATION In this section, we provide a proof of Theorem.. For the exact solution Φ of the nac equation., a detailed Taylor expansion, combined with a careful Fourier analysis, indicates the following consistency estimate: Φ k+ Φ k = Ms η Φ k, Φ k+ + B c Φ k+ B e Φ k+ [ J Φ k+ ] + sτ k+, 3. with the local truncation error with Φ k+ = τk+ i,j and the constant C 5 depends only on T, L,andL. We consider the following error function, at a pointwise level: In turn, subtracting.30 from 3. leads to Φ k +Φ k+, Φ k+ = 3 Φk Φk, 3. C 5 h + s, i, j, k, 3.3 e k i,j = Φk i,j φk i,j. 3.4 e k+ e k = Ms η Φ k, Φ k+ η φ k,φ k+ + B c e k+ B e ê k+ [ J ê k+ ] + sτ k i,j, with e k+ = e k + e k+, ê k+ = 3 ek ek. Taking a discrete inner-product with 3.5 by e k+/ =e k+ + e k, summing over i and j implies that e k+ ek + B cs e k+ + h s ηφ k, Φ k+ ηφ k,φ k+ e k+ sh τ k+ e k+ + B e sh ê k+ e k+ + h s [J ê k+ ] e k The term associated with the local truncation error could be bounded with an application of Cauchy inequality: h τ k+ e k+ τ k+ + ek+ τk+ + ek+ + ek. 3.7 The concave term could be bounded in a straightforward way: B e h ê k+ e k+ B e ê k+ + ek+ B e ek+ + 0 ek + ek. 3.8

10 GUAN ET AL The term associated with the convolution could be analyzed with the help of.: [J h ê k+ ] e k+ B J ê k+ + ek+ B 3.9 J ek+ + 0 ek + ek, with B J = [J c ] + [J e ]. The rest work is focused on the analysis for the term associated with the nonlinear error. We begin with the following decomposition: ηφ k, Φ k+ ηφ k,φ k+ = + + 3, with = φk+ +φ k e k+, = 4 Φk+ + φ k+ Φ k+ +Φ k e k+, = 4 Φk + φ k Φ k+ +Φ k e k. The following estimate is available for the term associated with : For the term associated with,wehave h e k+ h φ k+ e k+ h φ k e k+. 3. h 4 Φk+ Φ k+ +Φ k e k+ e k+ Φk+ Φ k+ + Φ k e k+ e k+ C e k+ e k+ C e k+ + ek, 3. h 4 φk+ Φ k+ +Φ k e k+ e k+ = h Φ k+ +Φ k e k+ φ k+ e k+ Φk+ + Φ k e k+ φ k+ e k+ C e k+ φ k+ e k+ C e k+ + φk+ e k+ = C e k+ + h φ k+ e k+, 3.3 with the Cauchy inequality applied in the last step. Consequently, we arrive at h e k+ C e k+ + ek +h φ k+ e k The estimate for the term associated with 3 is similar; the details are left to the interested readers: h 3 e k+ C e k+ + ek +h φ k e k In turn, a combination of 3., 3.4, and 3.5 into 3.0 yields the inner-product associated with the nonlinear error: h ηφ k, Φ k+ ηφ k,φ k+ e k+ C e k+ + ek. 3.6 Finally, a substitution of 3.7 to 3.9 and 3.6 into 3.6 indicates that e k+ ek + B cs e k+ Cs ek+ + ek + ek +s τk The application of discrete Gronwall inequality implies the l 0, T; l convergence estimate.4, using the local truncation error bound 3.3. The proof of Theorem. is complete. Remark 3.. In the convergence proof for Theorem., the decomposition 3.0 has played a key role in the nonlinear error estimates. Because of the well-posed nonlinear inner-product in 3., the degree of nonlinearity of the other nonlinear inner-products could be perfectly controlled, with only the maximum norm bound of the exact solution Φ needed. As a result of this technique, an estimate for the maximum norm of the numerical solution is avoided, which usually has to be obtained in the nonlinear convergence analysis. Because of this fact, an inverse inequality is not needed in the presented

11 6846 GUAN ET AL. analysis, and the l 0, T; l convergence for the nac equation turns out to be unconditional, ie, no scaling law between the time step size s and the grid size h is required for the desired convergence result. Remark 3.. We also observe that the technical assumption.48 for the physical parameters is not required in the convergence analysis for the nac equation. 4 HIGHER ORDER CONSISTENCY ANALYSIS OF.3 FOR THE NCH EQUATION: ASYMPTOTIC EXPANSION OF THE NUMERICAL SOLUTION For simplicity of presentation, we denote Φ k+ = Φ k + Φ k+, Φ k+ = 3 Φ k Φ k. 4. By consistency, the IPDE solution Φ solves the discrete equation Φ k+ Φ k = sδ h η Φ k, Φ k+ + B c Φ k+ B e Φ k+ where the local truncation error τ k+ satisfies τk+ i,j [ J Φ k+ ] + sτ k+, 4. C 6 h + s, 4.3 for all i, j, andk for some C 6 0 that depends only on T, L,andL. Meanwhile, it is observed that the leading local truncation error in 4. will not be enough to recover an a priori W, bound for the numerical solution, needed in the stability and convergence analysis. To remedy this, we use a higher order consistency analysis, via a perturbation argument, to recover such a bound in later analysis. In more detail, we need to construct supplementary fields, Φ h, Φ s,and Φ, satisfying Φ = Φ+h Φ h, + s Φ s,, 4.4 so that a higher Os 3 + h 4 consistency is satisfied with the given numerical scheme.3. The constructed fields Φ h,, Φ s,, which will be found using a perturbation expansion, will depend solely on the exact solution Φ. The following truncation error analysis for the spatial discretization can be obtained by using a straightforward Taylor expansion for the exact solution: t Φ =Δ h Φ 3 + [J ] + γ c γ e Φ [ J Φ ] + h g 0 + Oh 4, i, j. 4.5 Here the spatially discrete function g 0 is smooth enough in the sense that its discrete derivatives are bounded. Also note that there is no Oh 3 truncation error term, because the centered difference used in the spatial discretization gives local truncation errors with only even order terms, Oh, Oh 4,etc. The spatial correction function Φ h, is given by solving the following equation: t Φ h, =Δ h 3 Φ Φ h, + [J ] + γ c γ e Φ h, [ J Φ h, ] g 0, i, j. 4.6 Existence of a solution of the above linear system of ODEs is a standard exercise. Note that the solution depends only on the exact solution, Φ. In addition, the divided differences of Φ h, of various orders are bounded. Now, we define Φ h = Φ+h Φ h,. 4.7 A combination of 4.5 and 4.6 leads to the fourth-order local truncation error for Φ h : t Φ h =Δ h Φ h 3 + [J ] + γc γ e Φ h [ J Φ h] + Oh 4, i, j, 4.8 for which the following estimate was used: Φ h 3 = Φ+h Φ h, 3 = Φ 3 + 3h Φ Φ h, + Oh

12 GUAN ET AL We remark that the above derivation is valid since all Oh terms cancel in the expansion. Regarding the temporal correction term, we observe that the application of the second-order convex splitting scheme.3 for the profile Φ h gives Φ k+ h Φ h s k =Δ h η Φ h k, Φ h k+ + B c Φ k+ h Be Φ k+ h [ J Φ ] + s hk+ h + Os 3 +Oh 4, i, j, with Φ k+ Φ h = k h + Φ k+ h, Φ k+ 3 h = Φ h k 4.0 Φ k, h at any grid point i, j. In turn, the first-order temporal correction function Φ s, is given by the solution of the following system of linearized ordinary differential equations t Φ s, =Δ h 3 Φ h Φs, + [J c ] [J e ] + γ c γ e Φ s, [ J Φ s, ] h. 4. Again, the solution of 4., which exists and is unique, depends solely on the profile Φ and is smooth enough in the sense h that its divided differences of various orders are bounded. Similar to 4.0, an application of the second-order convex splitting scheme to Φ s, reads Φs, k+ Φs, k s with =Δ h + 4 Therefore, a combination of 4.0 and 4. shows that Φ k+ Φ k s Φ h k +Φ h k+ Φ h k Φ k s, +Φ h k+ Φ k+ s, Φ Φ k s, +Φk+ s, h k + Φ +B c Φ k+ s, B k+ eφ s, [J Φ ] k+ s, h k+ h k + sh k + Os +Osh, i, j, Φ k+ = Φ k s, s, +Φk+ k+, Φ 3 s, s, = Φk s, Φk s,. =Δ h η Φ k, Φ k+ + B c Φ k+ B e Φ k+ [ J Φ k+ ] + Os 3 + s 3 h + h 4, i, j, with Φ k+ = Φ k + Φ k+, Φ k+ = 3 Φ k Φ k, in which the construction 4.4 for the approximate solution Φ is recalled and we have used the following estimate η Φ k, Φ k+ = η Φ h k + s Φ k s,, Φ h k+ + s Φ k+ s, = η Φ h k, Φ h k+ + s Φ h k +Φ h k+ Φ h k Φ k s, +Φ h k+ Φ k+ s, + Φ 4 s Φ k s, +Φk+ h k + Φ h k+ + Os 4. s, 4.4 Remark 4.. Trivial initial data Φ h,, t = 0 0, Φ s,, t = 0 0aregiventoΦ h, and Φ s, as 4.6 and 4., respectively. Thus, we conclude that φ 0 Φ 0, φ k Φ k = 0, k These properties will be used in later analysis.

13 6848 GUAN ET AL. Remark 4.. The reason for such a higher order asymptotic expansion and truncation error estimate is to justify an a priori W, bound of the numerical solution, which is needed in the H convergence analysis. In more detail, in the expansion for the discrete gradient of the nonlinear error term and its H inner-product estimate with the error function, an a priori W, bound of the numerical solution at a discrete level is needed to control the l error of the numerical solution. To obtain such a bound, we have to perform a higher order consistency analysis, up to Os 3 +h 4, instead of the leading Os +h consistency for the exact solution, through a construction of approximate solution that satisfies the numerical scheme with higher order accuracy. In turn, an Os 3 + h 4 convergence in l 0, T; l norm can be derived, between the numerical solution and the constructed approximate solution, and this convergence result is used to recover the a priori W, assumption for the numerical solution, under a mild linear refinement constraint, s Ch, with the help of inverse inequality. 5 CONVERGENCE PROOF FOR THE NCH EQUATION As stated earlier, the purpose of the higher order expansion 4.4 is to obtain a W, bound of the error function via its L norm in higher order accuracy by using an inverse inequality in spatial discretization, which will be shown below. A detailed analysis shows that Φ Φ + h Φ Φ Cs + h, 5. since Φ h,, h Φ h,, Φ s,, h Φ s, C. Subsequently, the following error function is considered: ě k i,j = Φ k i,j φk i,j. 5. In other words, instead of a direct comparison between the numerical solution φ and the exact solution Φ or Φ, we estimate the error between the numerical solution and the constructed solution to obtain a higher order convergence in the norm, which follows the technique originally proposed in Strang. 63 Subtracting.3 from 4.3 yields ě k+ ě k = sδ h η Φ k, Φ k+ η φ k,φ k+ + B c ě k+ B e ě k+ [ J ě k+ ] + s τ k i,j, τ k i,j Cs3 + h 4, with ě k+ = + ě k+, ě k+ = ěk 3 ěk. 5. Preliminary error estimates for linear terms Proposition 5.. We have [ h J ě k+ ] Δ h ě k+ C 7 ě k α + ě k α h ě k+, α >0, 5.4 h ě k+ Δ h ě k+ h ě k+ h ě k + 5 h ě k+ + h ě k. 5.5 Proof. The first inequality 5.4 is a direct application of Lemma. and Cauchy inequality: [ h J ě k+ ] Δ h ě k+ sc α ě k+ + α h ě k+ C 3 ě k α + ě k + α 5.6 h ě k+. For the second inequality 5.5, we start from the summation by parts: h ě k+ Δ h ě k+ = h h ě k+ h ě k+ = h h ě k+ ěk h + ě k+. 5.7

14 GUAN ET AL Meanwhile, the term ě k+ can be rewritten as ě k+ = 3 ěk ěk = ě k+ ě k + ě k+ ě k, 5.8 which in turn gives the following estimate: h h ě k+ h ě k+ = h ěk+ h ě k ěk+ h + ě k + 4 h ě k+ h h ě k+ h ě k = h ě k+ h ě k + 4 h ě k+ h h ě k+ h ě k h ě k+ h ě k + 5 h ě k+ + h ě k. 5.9 Finally, its combination with 5.7 results in 5.5. The proof of Proposition 5. is complete. 5. Preliminary nonlinear error estimates The W, bound for the constructed approximate solution Φ is guaranteed by the regularity of the exact solution Φ and henceforth Φ and the correction terms Φ h,, Φ s,, at any time step. Similarly, its divided difference in time is also bounded pointwise, which comes from the regularity in time for the constructed solution. For the numerical solution φ, its global in time l 4 has been derived in Theorem.0. Moreover, to carry out the error estimate for the nonlinear term, we need to make an a priori W, assumption for the numerical solution at time step t k and use the Os 3 + h 4 order convergence in l to recover such an assumption at the next time step t k+. Proposition 5.. Suppose φ j, Φ j m n, are periodic with equal means, ie, φ j Φ j = 0,j= k, k +, and satisfying Φ k 4 + Φ k + h Φ k + Φ k+ 4 + Φ k+ + h Φ k+ C 0, 5.0 Φ k+ Φ k s φk 4 C 0, C 0, 5. φk+ 4 C 0, 5. φk + h φ k C 0, 5.3 where C 0 is an s, h-independent positive constant. Then, there exists a positive constant C, which depends on C 0 but is independent of s and h, such that h η Φ k, Φ k+ η φ k,φ k+ Δ h ě k+ C ě k α + ě k+ + α 3 h ě k C α s h ě k + C + φ k+ ě k, α >0. α Proof. For simplicity of presentation, we denote k = η Φ k, Φ k+ η φ k,φ k+. A direct application of summation bypartsrevealsthat h k Δ h ě k+ = h h k h ě k+ = k + k, with k = h D x k D x ě k+, k = h D y k 5.5 D y ě k+.

15 6850 GUAN ET AL. We focus on the first term k ; the second term k can be analyzed in the same way. In x direction, we drop the subscript j in the grid index, just for simplicity of presentation. A detailed expansion shows that D x η φ k,φ k+ i+ = φ k+ 4h i+ +φ k i+ φ k+ i+ + φk i+ φ k+ +φ k i i φ k+ + φ k i i = 4h + 4h = φ k+ i+ +φ k i+ +φ k+ i+ + φk+ φ k+ + φ k i i i φ k+ i+ φk+ + φ k i i+ φk i φ k i+ φk+ i+ + φk i φ k+ i φ k+ i + φ k i φ k i+ φk i φ k+ i+ +φ k i+ +φ k+ i+ + φk+ φ k+ + φ k i i i D x φ k+ i φ k i+ φk+ i+ + φk i φ k+ i φ k+ + φ k i i Dx φ k i+. A similar expansion can be made for D x η Φ k, Φ k+. In turn, we arrive at i+ D x η Φ k, Φ k+ η φ k,φ k+ = i+ k, + k, + k,3 + k,4, k, = φ k+ i+ +φ k i+ +φ k+ i+ + φk+ φ k+ + φ k i i i D x ě k+ i+, 5.7 k, = Φ k+ i+ + φk+ ěk+ i+ i+ + Φ k+ + φ k+ ěk+ i i i 5.8 +ě k+ i+ + ěk+ φ k+ + φ k k+ i i i + Φ i+ + Φ k+ ě k+ + ě k i i i D x Φ k+ i+, k,3 = 4 Φ k i+ Φ k+ i+ + Φ k i Φ k+ Φ k+ + Φ k i i i Dx ě k i+, 5.9 k,4 = ěk i+ 4 ěk+ i+ + ěk i ě k+ Φ k+ + Φ k i i i + φ k i+ φk+ i+ + φk i φ k+ ěk+ + ě k 5.0 i i i Dx φ k i+. For the term k, we observe the following estimate for the nonlinear coefficient:, =φ k+ i+ +φ k i+ +φ k+ i+ + φk+ φ k+ + φ k i i i =φ k+ i+ +φ k i+ + φ k+ i+ φk+ +φ k+ + φ k+ i i i+ φk i + φ k+ i φ k+ i+ +φ k+ +φ k i i+ + φ k+ i+ φk i + φ k+ φ k i i φ k i+ φ k i, φ k i 5. with a repeated application of Cauchy inequality in the last steps. Meanwhile, the a priori assumption 5.3 for the numerical solution φ at time step t k indicates that φ k i+ + φk i φ k i+ φk i = h φ k i+ + φk i D x φ k i+ h φ k h φ k C 0h, 5. at a pointwise level. As a result, its combination with 5.7 implies that h k, D xě k+ C 0 h D x ě k Similar estimates can be derived for k. The regularity assumption 5.0, 5. for the constructed approximate,3 solution Φ shows that Φ k i+ Φ k+ i+ + Φ k i Φ k+ i s Φ k+ Φ k s C 0 s, 5.4

16 GUAN ET AL. 685 Φ k+ i + Φ k i Φ k+ + Φ k C 0, so that 5.5 at a pointwise level. In turn, we arrive at 3 = Φ k i+ Φ k+ i+ + Φ k i Φ k+ Φ k+ + Φ k i i i 4C 0s, 5.6 h k,3 D xě k+ C 0 s D x ě k+ D x ě k For the second nonlinear term k, we start from a rewritten form:, 8 α 5.7 D x ě k+ + 8C4 0 α s D x ě k. k, = Φ k+ i+ + φk+ = i+ + φk+ i ě k+ i + φ k i ěk+ i+ + Φ k+ + φ k+ + φ k i i i ěk+ i D x Φ k+ i+, + Φ k+ i+ + Φ k+ i, ě k+ + i+, ě k+ + i,3ě k i+ +,4ě k i D x Φ k+ i+, with, =,3 = Φ k+ i+ + φk+ i+ + φk+ + φ k i i,,4 = Φ k+ + φ k+ + φ k i i i,, = Φ k+ + Φ k+ i i+ + φk+ + φ k i i. For these nonlinear coefficients, it is clear that 5.8, 4 +, 4 C Φ k 4 + Φ k+ 4 + φ k 4 + φ k+ 4 CC 0, 5.9,3 +,4 C Φ k + Φ k+ + φ k + φ k+ C C 0 + φ k+, 5.30 in which the regularity condition 5.0 and a priori assumption 5. and 5.3 were repeated used in the derivation. In particular, we note that the 4 bound is available for both the approximate solution Φ and the numerical solution φ, at both time steps t k and t k+, and the same for the bound for Φ. Meanwhile, in norm for the numerical solution φ, we only have its bound at time step t k, as an a priori assumption, and its bound at the next time step t k+ has to be obtained by a higher order convergence in l norm via an inverse inequality, as will be shown later. As a result, an application of discrete Hölder inequality shows that h k, D xě k+, 4 +, 4 ě k+ 4 +,3 +,4 ě k D x ě k+ 5.3 CC 0 ě k+ 4 + C C 0 + φ k+ ě k D x ě k+. Furthermore, a discrete Sobolev embedding in D gives 4 C ě k+ ěk+ h ě k+, since ě k+ = 0, 5.3 with C independent on h; its proof can be found in Guan et al. 48 We note that the zero-mean property of ě k+ comes from 4.5. Therefore, the first part in 5.3 can be bounded by CC 0 ě k+ 4 D x ě k+ M ě k+ 3 h ě k+ with M = CC In addition, we use the Young inequality a b ap p + bq q, a, b > 0, p + =, 5.34 q

17 685 GUAN ET AL. with the choices p = 4, q = 4, a = 6α 3 4 M 3 ě k+ 3, b = α 4 6 h ě k+ 3,andget CC 0 ě k+ 4 D x ě k+ M ě k+ 3 h ě k+ = a b 4 a b 4 3 = 4 M4 63 α 3 ě k+ + α 8 hě k The bound for the second part in 5.3 can be obtained by Cauchy inequality: C C 0 + C C φ k+ ě k D x ě k+ 0 + φk+ α ěk 8 α D x ě k Consequently, a combination of 5.3, 5.35, and 5.36 yields h k, D xě k+ C 8 ě k α + ě k+ + C φk+ 3 α + α h ě k+ 8 + D x ě k+, ěk with C 8 = 7M 4 + CC 0. The analysis for the fourth nonlinear term k,4 is similar to that of k. Its rewritten form reads as follows:, 5.37 k,4 = ěk i+ 4 ěk+ i+ + ěk i ě k+ Φ k+ i i = ěk+ i + Φ k i D x φ k i+, + φ k i+ φk+ i+ + φk i φ k+ i 4, ě k+ + i+ 4, ě k+ + i 4,3 ě k i+ + 4,4ě k i D x Φ k+ i+, 5.38 with 4, = Φ k+ + Φ k i i, 4, = 4,3 = 4,4 = Φ k+ + Φ k i i. Similarly, these nonlinear coefficients can be bounded by φ k i+ φk+ i+ + φk i φ k+ i Φ k+ i Φ k i, 4, 4 + 4, 4 C Φ k 4 + Φ k+ 4 + φ k 4 + φ k+ 4 CC 0, ,3 + 4,4 C Φ k + Φ k+ CC Note that for 4,3 and 4,4, since the numerical solution φ is not involved, the regularity assumption 5.0, 5. for the approximate solution Φ directly gives a bounded for these coefficients. This also greatly simplifies the analysis below. Then we have h k,4 D xě k+ 4, 4 + 4, 4 ě k ,3 + 4,4 ě k D x ě k+ CC 0 ě k+ 4 + ě k D x ě k+. In turn, the estimates 5.3 to 5.36 are also valid; consequently, the following estimate can be derived: h k,4 D xě k+ C 9 ě k α + ě k+ 3 + α h ě k+ 8 + D x ě k

18 GUAN ET AL Finally, a combination of 5.3, 5.7, 5.37, and 5.4 reveals that k = h D x k D x ě k+ C 0 α 3 ě k + ě k+ + C φ k+ α ěk + α 4 hě k+ + α D xě k+ + 8C4 0 α s D x ě k, by choosing h with C 0 h α. The analysis for 8 k is essentially the same: k = h D y k D y ě k+ C 0 α 3 ě k + ě k+ + C φk+ α ěk + α 4 hě k+ + α D yě k+ + 8C4 0 α s D y ě k ; and the details are skipped for brevity of presentation. Therefore, a combination of 5.43 and 5.44 results in 5.4. The proof of Proposition 5. is complete. Remark 5.3. In fact, for the nonlinear error term, the form of expansion and decomposition in its discrete gradient is not unique. However, the way in our decomposition 5.7 to 5.0 greatly facilitates the convergence analysis. It is well known that the exact solution Φ and the nonlinear potential Φ 3 have a nonpositive H inner-product, since 3Φ 0. However, for the second-order numerical approximation η φ k,φ k+, introduced by., its error estimate becomes much more tricky. In the decomposition 5.7, the nonlinear coefficient is proven to be almost nonnegative, as in 5., and the remainder term has an Oh bound given by 5., using the W, bound assumption for the numerical solution at t k, as given by 5.3. This treatment assures a controlled property of the nonlinear inner-product associated with 5.7. Moreover, since the numerical solution φ is involved with the nonlinear coefficient in 5.7, we could take the discrete gradient of the approximate solution Φ in the nonlinear expansion 5.8, and its norm is directly bounded by 5.0. If it is replaced by the discrete gradient of the numerical solution, a numerical analysis is not feasible, since a bound for φk+ is not available at time step t k+. Meanwhile, in the nonlinear expansion 5.0, an appearance of the discrete gradient of the numerical solution at time step t k does not cause any theoretical trouble, since we have had an a priori bound 5.3, which is to be recovered by an Os 3 + h 4 convergence analysis in l norm. For the nonlinear errors appearing in 5.8 and 5.0, we have to rewrite them in terms of a nonlinear combination of ě k+ and ě k. The reason is that we only have a well-posed diffusion term of hě k+ ; a positive diffusion term in either the form of hě k or hě k+ is not available in the numerical analysis, because of the second-order numerical approximation. With such a rewriting, the terms involving ě k+ only require an l 4 bound for the numerical and approximate solutions, given by 5.9, and the l 4 estimate for ě k+ is obtained by 5.3, a discrete Sobolev embedding. In turn, these terms can be controlled with the help of Young inequality, as in The terms involving ě k can be handled by a standard Cauchy inequality, and a coefficient φ k+ has to be included in the estimate Such a bound is not available at present; it has to be obtained from a preliminary estimate before a discrete Gronwall inequality is applied; see the analysis in later subsections. For the nonlinear expansion in 5.9, we make the nonlinear coefficient of order Os, as analyzed by 5.4 to 5.6. In addition, such a nonlinear coefficient has to be Φ dependent, instead of φ dependent, since we have not had the divided difference bound in time for the numerical solution. With such an Os analysis, the nonlinear inner-product associated with 5.9 is bounded by 5.7, in which the first part can be controlled by the diffusion term and the second part is an Os increment. The stability of such an Os incremental term is ensured by the term hě k+ hě k, which appears in 5.5 in Proposition 5., the estimate of the concave diffusion term. Remark 5.4. For the 3D case, a discrete Sobolev embedding gives 4 C ě k+ 4 3 ěk+ h ě k+ 4, if ě k+ = 0, 5.45

19 6854 GUAN ET AL. which is analogous to 5.3 in D; also see the related discussions in Guan et al. 48 In turn, we are able to derive the following result h 3 η Φ k, Φ k+ η φ k,φ k+ Δ h ě k+ ě k C α 7 + ě k+ + C α s h ě k + C α + α h ě k+ + φ k+ ě k, α >0, the only changes being the α 7 replaces α 3 andweusethetriplesummation. As a result, a full order convergence in 3D can be derived in the same manner. The details are omitted in this paper for the sake of brevity Proof of Theorem.: l 0, T; l convergence We begin with an Os 3 + h 4 convergence assumption of the numerical solution, in l norm, up to time step t k : ě j C e C t j s 3 + h 4, 0 j k, 5.47 with C, C independent on s and h. Consequently, an application of inverse inequality shows that ě j + h ě j C ě j CC ectj s3 + h4 Ch, 0 j k, 5.48 h d + h d + with the dimension d = or 3. It is also noted that the linear refinement constraint, s Ch, is used in the above derivation. In turn, the a priori assumption 5.3 for the numerical solution at t k is valid by setting C 0 =max Φ j + 0 j k h Φ j Moreover, it is clear that an estimate for φ k+ is needed in the application of Proposition 5. in the nonlinear analysis. For this quantity, we observe that 5., which comes from a global in time l 4 bound for the numerical solution as derived in Theorem.0, implies that φk+ C φk+ 4 CC 0 h d h d 4, with d the dimension, in which the first step comes from a similar inverse inequality. Now, we derive the l convergence at time step t k+. Multiplying by h ě k+ = h ě k+ + ě k, summing over i and j, and applying Green's second identity Proposition A.3, we have ě k + B ěk+ cs h ě k+ = h s η Φ k, Φ k+ η φ k,φ k+ Δ h ě k+ + sh τ k ě k+ B e sh ě k+ Δ h ě k+ h s [J ě k+ ] Δ h ě k+. Applying Propositions 5. and 5. for linear and nonlinear errors, and using the Cauchy inequality to bound the truncation error term h s τ k ě k+ sc 3 s 3 + h 4 + s ě k+ sc 3 s 3 + h 4 s + ě k+ + ě k, 5.5 we arrive at ě k + s B ěk+ c 5B e α h ě k+ + B es h ě k+ h ě k C α s ě k + ě k+ + C 4 3 α s ě k + ě k + B e s h ě k C α s3 h ě k + C α s φ k+ + sc ěk 3 s 3 + h 4, α >0. 5.5

20 GUAN ET AL A preliminary estimate for φk+ Note that an O bound for φk+ is not available at this point, because of the lack of information of the numerical solution at time step t k+. We only have 5.50, which comes from an unconditional l 4 stability of the numerical solution, and this bound may become singular as h 0. Meanwhile, such a bound is needed to apply the Gronwall inequality. To overcome this difficulty, we derive an estimate, based on 5.53, the assumption 5.47 up to time step t k, and the preliminary bound The assumption 5.47 implies that ěk Ch6, hě k, h ě k Ch4, φk+ Ch d, 5.54 with the standard constraint s Ch. Furthermore, using the fact that α 0 = B c 3B e > 0 and taking α = α 0, we conclude from 5.53 that C h 5 + h 7 d ěk+ Ch 5, since 7 d > 5ford =, In turn, an application of inverse inequality shows that C ěk+ Ch 5 d ěk+ h d Ch, with d = or Consequently, the triangular inequality yields φk+ Φ k+ + ě k+ C 5 = C Remark 5.5. Of course, the rough estimate 5.55 is not the convergence result that we want. Not only its accuracy is not satisfactory, Oh 5 instead of Os 3 + h 4, but also its stability is not maintained: Os 3 + h 4 convergence at the previous timesteptoanorderoh 5 at the next time step. The reason for such an accuracy deterioration is due to the singular bound 5.50 for φk+, which comes from the global in time l4 bound for the numerical solution. The purpose of the rough estimate 5.55 is to derive a preliminary convergence result in the l norm, based on the full convergence result at the previous time step, combined with the singular bound 5.50, so that a regular O bound can be obtained for the norm of the numerical solution at the next time step with an application of inverse inequality. Subsequently, the full order l convergence at the next time step can be derived by using the discrete Gronwall inequality, since an O bound for φk+ has been available l 0, T; l convergence and a recovery of the assumption 5.47 A substitution of 5.57 into 5.53 gives ě k ěk+ C 6 α 3 s + B es h ě k+ + s B c 5B e α h ě k+ ě k + ě k+ + C 4 α s ě k + B es h ě k s 3 + h 4, α >0. + C α s3 h ě k + sc 3 h ě k Replacing the index k by l, summing on l, froml = 0tol = k, and using ě 0 0 by 4.5, we have B es ěk+ h ě k+ + s α 0 α C6 s α + C k 4 k 3 α + sc 6 ěl α 3 l= l=0 k hě l l= + C ěl+ α s3 k hě l + sc 3 l=0 k h 4 + s 3, with B c 3B e = α 0 > 0 as in.48. As a direct consequence, by taking α = α 0, the following inequality holds: C 7 s + B es ěk+ h ě k+ sc 8 α 3 0 k el + C l= α s3 l=0 k hě l + sc 3 l= k h 4 + s 3, 5.60 l=0

21 6856 GUAN ET AL. with C 7 = C 6 α 3 we get and C 8 = C 6 + C 4 α. We can always choose s with C 7 s. In turn, by denoting G l = ě l + B es h ě l, 5.6 G k+ s C 8 α 3 0 k G l + sc 3 l= k h 4 + s 3, 5.6 with the choice of s so that C s B α e. An application of the discrete Gronwall inequality yields the desired result: ěk+ Gk+ C 9 s 3 + h 4, so that ě k+ C 9 s 3 + h 4, 5.63 with C 9 independent on s and h. A more detailed exploration implies the structure of this constant: C 9 = C e C t k+.asa result, the a priori assumption 5.47 is recovered at the time step t k+ so that an Os 4 + h 4 convergence in l norm, between the numerical solution and the constructed approximate solution Φ, has been established, using an induction argument. Finally, the proof of Theorem. can be completed with the following application of triangle inequality: el = φ l Φ l φ l Φ l + Φ l Φ l C s + h, l s T, 5.64 in which the error estimate 5.63 and the analysis 5. for the constructed solution are used. Remark 5.6. The assumption.48 for the physical parameters is required in the convergence analysis for the nch equation. Such an assumption is necessary for the convex diffusion part to control the concave diffusion part, because of a subtle estimate 5.5. As a consequence of this inequality, the assumption B c > 3B e has to be made to make the convergence analysis pass through. In comparison, for the nac equation, this assumption is not required, as explained in Remark 3.. On the other hand, our extensive numerical experiments have implied that such an assumption only corresponds to a technical difficulty in the convergence analysis. For most practical computational models, the second-order convergence is well preserved as long as the positive-diffusivity condition.5 is valid. Remark 5.7. We note that the second-order l 0, T; l convergence for the nch equation is conditional, ie, under a mild linear refinement constraint, s Ch. In comparison, the l 0, T; l convergence for the nac equation is unconditional, as explained in Remark 3.. Such a subtle difference comes from the analysis techniques for the nonlinear inner-products. For the nac equation, the decomposition 3.0 has greatly facilitated the error estimates, and the maximum norm bound of the numerical solution is not needed in the derivation. However, for the nch equation, since the discrete H inner-product of ě k+ and the nonlinear error function have to be analyzed, we need to make an a priori assumption 5.47 at the previous time step, obtain a discrete W, bound of the numerical solution, and the l 0, T; l convergence estimate justifies the a priori assumption at the next time step. This process is further facilitated by the higher order consistency analysis presented in Section 4. l=0 5.4 Proof of Theorem.3: l 0, T; l convergence With the Os 3 + h 4 convergence result 5.63, in l norm, we apply the inverse inequality and get C ě k+ C C 9 s 3 + h 4 C 0 s + h, with d =, 5.65 ěk+ h d h d with the linear refinement constraint s Ch and C 0 = C C 9. For the 3D case, a higher order asymptotic expansion of the numerical solution has to be performed so that an Os 4 + h 4 consistency and convergence in l norm are obtained. The details are left to interested readers. Subsequently, by combining the l error estimate 5.65 and the analysis 5. for the constructed solution, we finish the proof of Theorem. with an application of triangle inequality: el = φ l Φ l φ l Φ l + Φ l Φ l C s + h, l s T. 5.66

22 GUAN ET AL Remark 5.8. In fact, in the a priori assumption 5.47, the detailed form of constants C and C has not been fixed; instead, these constants are fixed by inequality 5.63, with C 9 = C e C t k+. In more details, C and C depend on the constants C 3, C 4,, C 8. However, this argument does not cause the trouble of a cycling analysis, because of the subtle fact that the subsequent constants C 3,, C 8 do not depend on C and C. In fact, for the a priori assumption 5.47, we only require that C and C are bounded by a fixed constant, so that inequality 5.48 becomes valid. With the availability of 5.48, all the subsequent constants C 3,, C 8 do not depend on C and C at all. 5.5 The l 0, T; l convergence for the nac equation For the second-order convex splitting scheme.30 for the nac equation, the higher order consistency analysis could be performed in the same manner as in Section 4. In turn, an l 0, T; l convergence estimate with an improved order Os 3 + h 4 is expected, and an application of inverse inequality leads to a similar l 0, T; l convergence result as Theorem.3, under the linear refinement path constraint s Ch. The proof of the following theorem is skipped for brevity, and the details are left to interested readers. Theorem 5.9. Under the assumptions of Theorem., we also have optimal order convergence of the numerical solution of the scheme.30 in the l norm. Namely, if s and h are sufficiently small with the linear refinement path constraint s Ch, with C any fixed constant, we have el C h + s, 5.67 where C > 0 is independent of h and s. 6 NUMERICAL RESULTS In this section, we present a few numerical experiments, verifying the convergence results of the second-order schemes for the nch and nac equations. 6. Numerical convergence for the nch equation Here we discuss the numerical results for the nch equation. We present cases, based on the restriction proposed in Equation.48. These experiments verify the convergence rate in the l 0, T; l norm. We use a square domain Ω = 0.5, 0.5 with smooth, periodic initial data 0.5sinπx sinπx. The convolution kernel J istakentobe J = α exp x + x, 6. σ where σ = 0.05 and α = σ.weextendj periodically outside of Ω. The other parameters are γ c = 4andγ e = 0 in the first case, which yield α 0 = 4 π>0; and γ c = 0andγ e = in the second case, which yield α 0 = 3 π<0. The final time forthetestsisgivenbyt = 0.0. To verify the spatial convergence order, we fix s = 0 4 and compare the difference function e A with respect to h = 64, 8, 56, 5, and 04. Since we do not have the exact solution these are not easily obtained for nontrivial convolution kernels we are using the difference between results on successive coarse and fine grids for the numerical comparison. The difference function, e A, is evaluated at time T = 0.0 using the method described in previous studies. 48,58,59 To verify the temporal convergence order, we fix h = 04 and compare the difference function e A with respect to s = 0.00, , , , , and The result is displayed in Tables and α 0 = 4 π >0 and Tables 3 and 4 α 0 = 3 π <0. In both cases, the spatial and temporal second-order accuracy of the method is confirmed. 6. Numerical convergence for the nac equation Here we discuss the numerical results for the nac equation. First we present the experiment verifying the numerical convergence rate. The setting of the experiment is the same as the nch case, with γ c = 0andγ e = which yields α 0 = 3 π<0. The result is displayed in Tables 5 and 6. The spatial and temporal second-order accuracy of the method is confirmed.

23 6858 GUAN ET AL. TABLE The difference between coarse and fine spatial discretization of the computed numerical solutions, with α 0 = 4 π and s = Coarse h Fine h e A Rate /64 / e-05 /8 / e /56 / e /5 / e The second-order spatial convergence of the method is confirmed in the test. TABLE The difference between coarse and fine temporal discretization of the computed numerical solutions, with α 0 = 4 π and h = 04 Coarse s Fine s e A Rate The second-order temporal convergence of the method is confirmed in the test. TABLE 3 The difference between coarse and fine spatial discretization of the computed numerical solutions, with α 0 = 3 π and s = Coarse h Fine h e A Rate /64 / e-04 /8 / e /56 / e /5 / e The second-order spatial convergence of the method is confirmed in the test. TABLE 4 The difference between coarse and fine temporal discretization of the computed numerical solutions, with α 0 = 3 π and h = 04 Coarse s Fine s e A Rate The second-order temporal convergence of the method is confirmed in the test. TABLE 5 The difference between coarse and fine spatial discretization of the computed numerical solutions, with γ 0 = 3 π and s = Coarse h Fine h e A Rate /8 / e-04 /56 / e /5 / e /04 / e The second-order spatial convergence of the method is confirmed in the test. We also present experiments of phase separation described by the nac equation under the following conditions: Ω = 0, 0 ; the size of time step is s = 0.0, the number of nodes on grid is 5 and the total number of time iterations is

24 GUAN ET AL TABLE 6 The difference between coarse and fine temporal discretization of the computed numerical solutions, with α 0 = 3 π and h = 04 Coarse s Fine s e A Rate e e e e The second-order temporal convergence of the method is confirmed in the test. FIGURE Phase separation described by nonlocal Allen-Cahn equation [Colour figure can be viewed at wileyonlinelibrary.com] FIGURE Energy evolution of phase separation [Colour figure can be viewed at wileyonlinelibrary.com] 0 4 ; 3 The convolution kernel J is a function defined as the difference between Gaussians: J = α exp x + x β exp x + x, 6. σ σ where σ = 0.6, σ = 0.4, α = 0. σ and β = 0.08 ; 4γ σ e = 0, γ c = 0. The initial condition of the simulation is a random perturbation of the constant state φ ave = 0. Figure shows snapshots of the evolution up to time T = 00, and Figure shows the corresponding numerical energy for the simulation. The energy is observed to decay as time increases. 7 CONCLUDING REMARKS In this paper, we have presented the detailed convergence analyses for fully discrete second-order numerical schemes for nac and nch equations. For the nac equation, the standard procedure of consistency and stability estimates indicates the desired convergence result, without any constraint between time step s and spatial grid size h, in which a careful nonlinear expansion