Structure-preserving Numerical Schemes for Phase Field Models

In this thesis we study how to design accurate, efficient and structure-preserving numerical schemes for phase field models including the Allen–Cahn equation, the Cahn–Hilliard equation and the molecular beam epitaxy equation. These numerical schemes include the explicit Runge–Kutta methods, exponential time differencing (ETD) Runge–Kutta methods and implicit-explicit (IMEX) Runge–Kutta methods. Note that the phase field models under consideration are gradient flows whose energy functionals decrease with time. For the Allen–Cahn equation, it is well known that the solution satisfies the maximum principle; for the Cahn–Hilliard equation, although its solution does not satisfy the maximum principle, the solution is also bounded in time. When designing numerical schemes, we wish to preserve certain stabilities satisfied by the physical solutions. We first make use of strong stability preserving (SSP) Runge-Kutta methods and apply some detailed analysis to derive a class of high-order (up to 4) explicit Runge-Kutta methods which not only decrease the discrete energy but also preserve the maximum principle for the Allen–Cahn equation. Secondly, we prove that the second-order exponential time differencing Runge-Kutta methods decrease the discrete energy for the phase field equations under investigation. Moreover, it can be shown that the ETDRK methods can also preserve the maximum bound property for the Allen–Cahn equation. What is more important is that both properties are preserved unconditionally, in the sense that the stability conditions do not depend on the size of time steps. Although the proof is only valid for second-order schemes and still open for higher-order methods, its numerical efficiency has been well observed in computations. The third approach is the implicit-explicit (IMEX) Runge–Kutta (RK) schemes, i.e. taking the linear part in the equation implicitly and the nonlinear part explicitly. A class of high-order IMEX-RK schemes are studied carefully. We demonstrate that some of the IMEX-RK schemes can preserve the energy decreasing property unconditionally for all the phase-field models under investigation.

[1] S. M.AllenandJ.W.Cahn, A microscopictheoryforanti-phaseboundarymotionand itsapplicationtoanti-phasedomaincoarsening, ActaMetall,27(1979),pp.1085–1095.

[2] U. M.Ascher,S.J.Ruuth,andR.J.Spiteri, Implicit-explicit Runge-Kuttamethodsfortime-dependentpartialdifferentialequations, AppliedNumericalMathe-matics, 25(1997),pp.151–167.

[3] K. Burrage,W.H.Hundsdorfer,andJ.G.Verwer, A studyofB-convergenceof Runge-Kuttamethods, Computing,36(1986),p.17â34.

[4] J. C.Butcher, Numericalmethodsforordinarydifferentialequations, Wiley,(2003).

[5] J. H.Chaudhry,J.Collins,andJ.N.Shadid, A posteriorierrorestimationformulti-stage Runge-KuttaIMEXschemes, AppliedNumericalMathematics,117(2017),pp. 36–49.

[6] L. Chen, Phase-field modelsformicrostructuralevolution, Ann.Rev.Mater.Res.,32(2002), pp.113–140.

[7] L. ChenandJ.Shen, Applicationsofsemi-implicitFourier-spectralmethodtophasefield equations, Comput.Phys.Commun.,108(1998),pp.147–158.

[8] Q. ChengandJ.Shen, Multiple scalarauxiliaryvariable(MSAV)approachandits applicationtothephase-fieldvesiclemembranemodel, SIAMJournalonScientificComputing, 40(2018),pp.A3982–A4006.

[9] S. M.CoxandP.C.Matthews, Exponentialtimedifferencingforstiffsystems, J.Comput. Phys.,176(2002),pp.430–455.

[10] L. DongandQ.Zhonghua, On secondordersemi-implicitFourierspectralmethodsfor 2DCahn-Hilliardequations, J.Sci.Comp.,70(2017),pp.301–341.

[11] L. Dong,Q.Zhonghua,andT.Tao, Characterizingthestabilizationsizeforsemi-implicit Fourier-spectralmethodtophasefieldequations, SIAMJournalonNumericalAnalysis, 54(2016),pp.1653–1681.

[12] Q. Du,L.Ju,X.Li,andZ.Qiao, Maximum principlepreservingexponentialtimedifferencingschemesforthenonlocalAllen–Cahnequations, SIAMJ.Numer.Anal.,57 (2019),pp.875–898.

[13] Q. Du,L.Ju,X.Li,andZ.Qiao, Maximum boundprinciplesforaclassofsemi-linearparabolicequationsandexponentialtimedifferencingschemes, SIAMReview,63 (2021),pp.317–359.

[14] Q. DuandJ.Yang, AsymptoticallycompatibleFourierspectralapproximationsofnonlocalAllen–Cahnequations, SIAMJ.Numer.Anal.,54(2016),pp.1899–1919.

[15] Q. DuandW.Zhu, Stability analysisandapplicationoftheexponentialtimediffer-encing schemes, J.Comput.Math.,22(2004),pp.200–209.

[16] C. M.Elliott, The Cahn-Hilliardmodelforthekineticsofphaseseparation, Mathe-matical modelsforphasechangeproblems(Obidos,1988),Internat.Ser.Numer.Math.,vol.88,Birkhauser,Basel,88(1989),pp.35–73.

[17] C. M.ElliottandA.M.Stuart, The globaldynamicsofdiscretesemilinearparabolicequations, SIAMJ.Numer.Anal.,30(1993),pp.1622–1663.

[18] H. Emmerich, The diffuseinterfaceapproachinmaterialsscience, Springer,NewYork,(2003).

[19] D. J.Eyre, Anunconditionallystableone-stepschemeforgradientsystems, unpub-lished,http://www.math.utah.edu/eyre/research/methods/stable.ps.,(1997).

[20] D. J.Eyre, UnconditionallygradientstabletimemarchingtheCahn-Hilliardequation,Mater. Res.Soc.Symp.Proc.,529(1998),pp.39–46.

[21] X. Feng,T.Tang,andJ.Yang, Longtimenumericalsimulationsforphase-fieldproblemsusing p-adaptive spectraldeferredcorrectionmethods, SIAMJournalonSci-entificComputing,37(2015),pp.A271–A294.

[22] L. FerracinaandM.N.Spijker, Stepsize restrictionsforthetotal-variation-diminishing propertyingeneralRunge–Kuttamethods, SIAMJournalonNumericalAnalysis, 42(2004),pp.1073–1093.

[23] E. FriedandM.E.Gurtin, Dynamic solid-solidtransitionswithphasecharacterizedby anorderparameter, PhysicsD:NonlinearPhenomena,72(1994),pp.287–308.

[24] L. Golubovic,A.Levandovsky,andD.Moldovan, Interfacedynamicsandfar-from-equilibriumphasetransitionsinmultilayerepitaxialgrowthanderosiononcrystalsurfaces:Continuumtheoryinsights, EastAsianJ.Appl.Math.,1(2011),pp.297–371.

[25] H. Gomez,L.Cueto-Felgueroso,andR.Juanes, Three-dimensionalsimula-tion ofunstablegravity-driveninfiltrationofwaterintoaporousmedium, JournalofComputation Physics,238(2013),pp.217–239.

[26] H. GomezandT.Hughes, Provablyunconditionallystable,second-ordertime-accurate,mixedvariationalmethodsforphase-fieldmodels, J.Comput.Phys.,230(2011), pp.5310–5327.

[27] Y. Gong,Q.Wang,Y.Wang,andJ.Cai, A conservativeFourierpseudo-spectralmethodforthenonlinearSchrodingerequation, JournalofComputationalPhysics,328(2017), pp.354–370.

[28] C. Gottlieb,SamdShuandE.Tadmor, Strongstability-preservinghigh-ordertime discretizationmethods, SIAMRev.,43(2001),pp.89–112.

[29] S. Gottlieb,D.I.Ketcheson,andC.-W.Shu, StrongstabilitypreservingRunge–Kutta andmultisteptimediscretizations, WorldScientificPress,(2011).

[30] S. GottliebandC.-W.Shu, TotalvariationdiminishingRunge–Kuttaschemes,Mathematics ofComputation,67(1998),pp.73–85.

[31] Z. Guan,C.Wang,andS.Wise, A convergentconvexsplittingschemefortheperiodicnonlocalCahn–Hilliardequation, Numer.Math.,128(2014),pp.377–406.

[32] L. Ju,X.Li,Z.Qiao,andH.Zhang, Energystabilityanderrorestimatesofexpo-nential timedifferencingschemesfortheepitaxialgrowthmodelwithoutslopeselection,Math. Comp.,87(2018),pp.1859–1885.

[33] J. Kim, Phase-field modelsformulti-componentfluidflows, Commun.Comput.Phys.,12 (2012),pp.613–661.

[34] J. F.B.M.Kraaijevanger, ContractivityofRunge–Kuttamethods, BIT,31(1991),p. 482â528.

[35] L. Leibler, Theoryofmicrophaseseparationinblockcopolymers, Macromolecules,13(6) (1980),pp.1602–1617.

[36] B. LiandJ.Liu, Thin filmepitaxywithorwithoutslopeselection, EuropeanJ.Appl.Math., 14(2003),pp.713–743.

[37] J. Lili,L.Xiao,Q.Zhonghua,andJ.Yang, Maximum boundprinciplepreservingintegratingfactorRungeâKuttamethodsforsemilinearparabolicequations, J.Comp.Phys.,439(2021).

[38] O. PenroseandP.C.Fife, Thermodynamicallyconsistentmodelsofphase-fieldtypeforthekineticsofphasetransition, Phys.D,43(1990),pp.44–62.

[39] S. J.RuuthandR.J.Spiteri, Two barriersonstrong-stability-preservingtimediscretizationmethods, J.Sci.Comput.,17(2002),pp.211–220.

[40] J. Shen,T.Tang,andJ.Yang, On themaximumprinciplepreservingschemesforthe generalizedAllen–Cahnequation, Comm.Math.Sci.,14(2016),pp.1517–1534.

[41] J. Shen,J.Xu,andJ.Yang, The scalarauxiliaryvariable(SAV)approachforgradientflows, JournalofComputationalPhysics,353(2018),pp.407–416.

[42] J. Shen,J.Xu,andJ.Yang, A newclassofefficientandrobustenergystableschemes forgradientflows, SIAMRev.,61(2019),pp.474–506.

[43] J. ShenandX.Yang, NumericalapproximationsofAllen–CahnandCahn–Hilliardequations, Discret.Contin.Dyn.Syst.,28(2010),pp.1669–1691.

[44] C. Shu, Total-variation-diminishingtimediscretizations, SIAMJournalonScientificand StatisticalComputing,9(1988),pp.1073–1084.

[45] C.-W. ShuandS.Osher, Efficient implementationofessentiallynon-oscillatoryshock-capturingschemes, JournalofComputationalPhysics,77(1988),pp.439–471.

[46] T. Tang, Revisitofsemi-implicitschemesforphase-fieldequations, Anal.TheoryAppl., 36(3)(2020),pp.235–242.

[47] T. TangandJ.Yang, Implicit-explicit schemefortheAllen–Cahnequationpreservesthe maximumprinciple, J.Comput.Math.,34(2016),pp.451–461.

[48] J. D.vanderWaals, The thermodynamictheoryofcapillarityunderthehypothesisof acontinuousvariationofdensity, J.Stat.Phys.,20(1979),pp.197–244.

[49] J. G.Verwer, ConvergenceandorderreductionofdiagonallyimplicitRunge-Kuttaschemes inthemethodoflines, Proc.DundeeNumericalAnalysisConference,140(1985), pp.220–237.

[50] L. Xiao,Q.Zhonghua,andW.Cheng, Convergenceanalysisforastabilizedlinearsemi-implicit numericalschemeforthenonlocalCahnâHilliardequation, Math.Comp.,90 (2021),pp.171–188.

[51] C. XuandT.Tang, Stability analysisoflargetime-steppingmethodsforepitaxialgrowthmodels, SIAMJ.Numer.Anal.,44(2006),pp.1759–1779.

[52] J. Yang,Q.Du,andW.Zhang, Uniform lp -boundoftheAllen–Cahnequationandits numericaldiscretization, InternationalJournalofNumericalAnalysisandModeling,15 (1-2)(2018),pp.213–227.

[53] X. Yang, Linear,firstandsecond-order,unconditionallyenergystablenumericalschemes forthephasefieldmodelofhomopolymerblends, JournalofComputationalPhysics,327(2016),pp.294–316.

[54] X. Yang,J.Zhao,Q.Wang,andJ.Shen, Numericalapproximationsforathree-componentCahn–Hilliardphase-fieldmodelbasedontheinvariantenergyquadrati-zation method, MathematicalModelsandMethodsinAppliedSciences,27(2017),pp. 1993–2030.