整数阶和分数阶相场方程高精度稳定算法

本篇论文致力于研究并分析整数和分数阶相场方程高精度稳定算法.  相场模型在模拟界面问题时有着极大的优势. 近年来, 对于相场模型构造高效且能量稳定的数值算法非常活跃. 本文结合了标量辅助变量法以及间断有限元方法构造并分析了整数阶相场方程的高精度能量稳定的全离散数值算法. 

由于分数阶相场方程具有弱奇异性、非局部性以及多尺度行为,  直接研究分数阶相场方程的高精度稳定算法十分困难. 本文首先在一般非均匀网格上研究Caputo分数阶导数的L2-1$_\sigma$、L2和快速L2-1$_\sigma$ 离散算子的性质, 证明了关于以上三种算子所对应的相关双线性型的正定性.  这也是保证相应时间分数阶相场方程的离散格式能量稳定的关键.
其次把相关结果应用于时间分数阶扩散方程(次扩散方程), 得出相关数值格式的长时间$H^1$稳定性. 最后基于正定性结果, 对时间分数阶相场方程构造出高精度稳定算法. 

本文的内容如下:

第一章介绍问题的研究背景和现状,  以及本文的主要研究内容.  我们介绍了整数以及分数阶相场方程的背景, 及其相关研究的挑战性.

第二章对相场方程构造并分析了一种高精度稳定全离散格式, 该方法在时间和空间上分别使用带标量辅助变量的 Runge--Kutta (RK-SAV) 外推法和间断Galerkin (DG) 方法. 基于一种新的解耦技巧, 该格式只需 求解几个具有常系数的椭圆标量问题. 对于相场梯度流问题, 该算法保持离散能量递减.   此外本章证明了该格式对Allen-Cahn 和Cahn-Hilliard 方程 是时空任意高阶, 具体来说 对于$q$ 级RK--SAV/DG外推 算法 有 空间上 最优 $L^2$ 误差估计和时间上$q$ 阶收敛率. 

第三章关注时间分数阶问题的高阶数值算法.  本章首先在一般时间非均匀网格上分析Caputo导数的 L2-1$_\sigma$、 L2 以及快速L2-1$_\sigma$离散系数的单调性,  其次证明了当时间步长满足一定条件时,  L2-1$_\sigma$、L2 和快速L2-1$_\sigma$离散分数阶算子所对应的双线性型正定. 此外基于以上系数的分析以及正定性结果, 对次扩散方程的L2-1$_\sigma$、 L2 和快速L2-1$_\sigma$ 格式给出数值解的 全局$H^1$ 稳定性 以及初步收敛性分析. 具体来说在一般时间非均匀网格上, 放宽了次扩散方程的L2-1$_\sigma$ 和快速L2-1$_\sigma$ 格式最优误差分析步长比的限制条件以及 首次给出了次扩散方程的 L2格式次优的$H^1$-范数误差分析.  本章最后一节考虑困扰学者很长时间的导致错误数值模拟的舍入误差问题,  其原因是由于计算机中的消除灾难. 本章提出一种处理消除灾难的新框架, 特别是在一般时间非均匀网格上, 计算标准和快速 L2型方法的系数. 首先引进 $\delta$-消除的概念, 与此同时提出了一些阈值条件来确保 不会发生$\delta$-消除. 如果不满足阈值条件, 则用Taylor展开技巧以避免 $\delta$-消除. 

第四章基于上一章研究的分数阶离散算子的性质给出时间分数阶相场方程的高精度稳定算法. 首先对基于二次线性化的快速L2-1$_\sigma$格式, 结合正定性结果给出能量稳定的证明. 此外基于SAV方法和快速L2-1$_\sigma$近似, 对时间分数阶相场方程提出了一个无条件能量稳定的快速算法.

This thesis is dedicated to study and analysis of high-order stable methods for integer and fractional order phase-field equations. Phase-field approach has been demonstrated  successful  in simulating interface problems. Efficient and energy stable numerical schemes for phase-field equations are very prevalent in the last few decades. In this thesis, we will combine the    scalar auxiliary variable method  and discontinuous Galerkin (DG) method to obtain  fully discrete schemes for the phase-field  equations, which can be of arbitrarily high order and satisfying energy stability. 
    
    Due to the weak singularity, non-locality and multi-scale behaviors of the  time-fractional phase-field (TFPF) equations, it is very difficult to directly study  high-order stable schemes for the TFPF equations.  In this thesis, we first investigate the properties for  the L2-1$_\sigma$, L2 and fast L2-1$_\sigma$   discrete fractional-derivative operators on genneral nonuniform meshes.  Then the positive definiteness for the bilinear forms associated with the three operators is examined, which is the  core issue  in constructing the energy stable schemes for TFPF equations. Based on the positive definiteness results, the long time $H^1$-stabilities of high order methods are then derived for subdiffusion equations. Finally, some high-order stable methods for TFPF equations are provided.
    
     This thesis is organized as follows.
     
Chapter 1 presents the research background and current development,  as well as the main contents of this thesis.  We introduce the background of integer and fractional order phase field equations.  The difficulties in studying high order stable methods for the phase-field equations  are explained. 

Chapter 2  constructs and analyzes a stable higher order fully discrete method for phase-field equations,  which uses extrapolated Runge--Kutta with scalar auxiliary variable (RK-SAV) method in time and DG method in space.  A new technique is proposed to decouple the system,  after which only several elliptic scalar problems with constant coefficients need to be solved independently. Discrete energy diminishing property of the method is proved. The scheme can be of arbitrarily high order both in time and space,  which is demonstrated rigorously for the Allen--Cahn equation and the Cahn--Hilliard equation. More precisely,  optimal $L^2$-error bound in space and $q$th-order convergence rate in time are obtained for $q$-stage extrapolated RK-SAV/DG method. 

 Chapter 3 focus on high order methods for time-fractional problems.   The monotonicity of the L2-1$_\sigma$,  L2 and  fast L2-1$_\sigma$  coefficients for Caputo derivative is provided and proved first.  Then  under some mild restrictions on time stepsize, the positive definiteness of the bilinear forms associated with the L2-1$_\sigma$, L2  and  fast L2-1$_\sigma$  formulas is proved.   As a consequence,  the uniform global-in-time $H^1$-stability of the L2-1$_\sigma$,   L2 and   fast L2-1$_\sigma$ schemes can be  derived  for subdiffusion equations,  in the sense that the $H^1$-norm is uniformly bounded as the time tends to infinity. 
 In addition, the sharp error analysis is proved for the L2-1$_\sigma$  and  fast L2-1$_\sigma$  methods on general nonuniform meshes for subdiffusion equations, where the restriction on time step ratios is relaxed comparing to existing works. The sub-optimal $H^1$-norm error analysis is provied for L2  method of subdiffusion equations on general nonuniform meshes. To the best of our knowledge, this is the first work on the  $H^1$-norm convergence for L2 method on general nonuniform meshes.  In the end of this Chapter, the  roundoff error problems are investigated, which  have troubled researchers  for a long time. The  roundoff issue occurred frequently in interpolation methods of time-fractional equations,  which can lead to wrong results in simulations.  These problems are essentially caused by catastrophic cancellations. In this Chapter,  a new framework to handle catastrophic cancellations is proposed,  in particular,  in the computation of the coefficients for standard and fast L2-type methods on general nonuniform meshes. A concept of $\delta$-cancellation is proposed  with some threshold conditions which ensure that $\delta$-cancellations will not happen. 
If the threshold conditions are not satisfied,  then a Taylor-expansion technique is proposed to avoid $\delta$-cancellation. 
 
 In Chapter 4, based on the properties of fractional discrete operators studied in  Chapter 3, high-order stable numerical methods for TFPF equations are provieded. The fast L2-1$_\sigma$ scheme based on second order linearization is considered first. Then  a fast unconditional energy stable method  is constructed for TFPF equations by combining the SAV method and the fast L2-1$_\sigma$ approximation.

[1] Van der Waals J D. The thermodynamic theory of capillarity under the hypothesisof a continuous variation of density[J]. Journal of Statistical Physics, 1979, 20(2):200–244.
[2] Landau L. The theory of phase transitions[J]. Nature, 1936, 138(3498): 840–841.
[3] Langer J. Models of pattern formation in first-order phase transitions[M].[S.l.]:World Scientific, 1986: 165–186.
[4] Anderson D M, McFadden G B, Wheeler A A. Diffuse-interface methods in fluidmechanics[J]. Annual Review of Fluid Mechanics, 1998, 30(1): 139–165.
[5] Boettinger W J, Warren J A, Beckermann C, et al. Phase-field simulation of solidification[J]. Annual Review of Materials Research, 2002, 32(1): 163–194.
[6] Frieboes H B, Jin F, Chuang Y, et al. Three-dimensional multispecies nonlineartumor growth—II: tumor invasion and angiogenesis[J]. Journal of Theoretical Biology, 2010, 264(4): 1254–1278.
[7] Braun R, Murray B. Adaptive phase-field computations of dendritic crystalgrowth[J]. Journal of Crystal Growth, 1997, 174(1-4): 41–53.
[8] Villain J. Continuum models of crystal growth from atomic beams with and withoutdesorption[J]. Journal de Physique I, 1991, 1(1): 19–42.
[9] Wang Y, Jin Y, Khachaturyan A G. Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films[J]. Acta Materialia,2003, 51(14): 4209–4223.
[10] Allen S M, Cahn J W. A microscopic theory for antiphase boundary motion andits application to antiphase domain coarsening[J]. Acta Metallurgica, 1979, 27(6):1085–1095.
[11] Dobrosotskaya J A, Bertozzi A L. A wavelet-Laplace variational technique forimage deconvolution and inpainting[J]. IEEE Transactions on Image Processing,2008, 17(5): 657–663.
[12] Ilmanen T. Convergence of the Allen-Cahn equation to Brakke’s motion by meancurvature[J]. Journal of Differential Geometry, 1993, 38(2): 417–461.
[13] Feng X, Prohl A. Numerical analysis of the Allen-Cahn equation and approximationfor mean curvature flows[J]. Numerische Mathematik, 2003, 94: 33–65.
[14] Feng X, Wu H. A posteriori error estimates and an adaptive finite element methodfor the Allen–Cahn equation and the mean curvature flow[J]. Journal of ScientificComputing, 2005, 24: 121–146.
[15] Elliott C M, Stinner B. Computation of two-phase biomembranes with phase dependentmaterial parameters using surface finite elements[J]. Communications inComputational Physics, 2013, 13(2): 325–360.
[16] Golubović L, Levandovsky A, Moldovan D. Interface dynamics and far-fromequilibrium phase transitions in multilayer epitaxial growth and erosion on crystalsurfaces: Continuum theory insights[J]. East Asian Journal on Applied Mathematics, 2011, 1(4): 297–371.
[17] Kim J. Phase-field models for multi-component fluid flows[J]. Communicationsin Computational Physics, 2012, 12(3): 613–661.
[18] Cahn J W, Hilliard J E. Free energy of a nonuniform system. I. Interfacial freeenergy[J]. The Journal of Chemical Physics, 1958, 28(2): 258–267.
[19] Bertozzi A, Esedoglu S, Gillette A. Analysis of a two-scale Cahn–Hilliard modelfor binary image inpainting[J]. Multiscale Modeling and Simulation, 2007, 6(3):913–936.
[20] Bertozzi A, Esedoglu S, Gillette A. Inpainting of binary images using the Cahn–Hilliard equation[J]. IEEE Transactions on image processing, 2006, 16(1):285–291.
[21] Shen J, Yang X. Energy stable schemes for Cahn-Hilliard phase-field model of twophase incompressible flows[J]. Chinese Annals of Mathematics, Series B, 2010,31(5): 743–758.
[22] Shen J, Yang X. A phase-field model and its numerical approximation for two-phaseincompressible flows with different densities and viscosities[J]. SIAM Journal onScientific Computing, 2010, 32(3): 1159–1179.
[23] Adams E E, Gelhar L W. Field study of dispersion in a heterogeneous aquifer: 2.Spatial moments analysis[J]. Water Resources Research, 1992, 28(12): 3293–3307.
[24] Nigmatullin R. The realization of the generalized transfer equation in a mediumwith fractal geometry[J]. Physica Status Solidi (B), 1986, 133(1): 425–430.
[25] Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena[J]. Chaos, Solitons and Fractals, 1996, 7(9): 1461–1477.
[26] Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introductionto mathematical models[M].[S.l.]: World Scientific, 2022.
[27] Hatano Y, Hatano N. Dispersive transport of ions in column experiments: Anexplanation of long-tailed profiles[J]. Water Resources Research, 1998, 34(5):1027–1033.
[28] Solomon T, Weeks E R, Swinney H L. Observation of anomalous diffusion andLevy flights in a two-dimensional rotating flow[J]. Physical Review Letters, 1993,71(24): 3975.
[29] Metzler R, Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[J]. Journalof Physics A: Mathematical and General, 2004, 37(31): R161.
[30] Gorenflo R, Mainardi F, Moretti D, et al. Time fractional diffusion: a discreterandom walk approach[J]. Nonlinear Dynamics, 2002, 29(1): 129–143.
[31] Caputo M. Linear models of dissipation whose Q is almost frequency independent—II[J]. Geophysical Journal International, 1967, 13(5): 529–539.
[32] Liu H, Cheng A, Wang H, et al. Time-fractional Allen–Cahn and Cahn–Hilliardphase-field models and their numerical investigation[J]. Computers and Mathematics with Applications, 2018, 76(8): 1876–1892.
[33] Tang T, Yu H, Zhou T. On energy dissipation theory and numerical stability fortime-fractional phase-field equations[J]. SIAM Journal on Scientific Computing,2019, 41(6): A3757–A3778.
[34] Zhao J, Chen L, Wang H. On power law scaling dynamics for time-fractional phasefield models during coarsening[J]. Communications in Nonlinear Science and Numerical Simulation, 2019, 70: 257–270.
[35] Du Q, Yang J, Zhou Z. Time-fractional Allen–Cahn equations: analysis and numerical methods[J]. Journal of Scientific Computing, 2020, 85(2): 42.
[36] Li Z, Wang H, Yang D. A space–time fractional phase-field model with tunablesharpness and decay behavior and its efficient numerical simulation[J]. Journal ofComputational Physics, 2017, 347: 20–38.
[37] Quan C, Wang B. Energy stable L2 schemes for time-fractional phase-field equations[J]. Journal of Computational Physics, 2022, 458: 111085.
[38] Liao H, McLean W, Zhang J. A Second-Order Scheme with Nonuniform TimeSteps for a Linear Reaction-Subdiffusion Problem[J]. Communications in Computational Physics, 2021, 30(2): 567–601.
[39] Chen H, Stynes M. Error analysis of a second-order method on fitted meshes for atime-fractional diffusion problem[J]. Journal of Scientific Computing, 2019, 79(1):624–647.
[40] Eyre D J. Unconditionally gradient stable time marching the Cahn-Hilliard equation[J]. MRS Online Proceedings Library (OPL), 1998, 529: 39.
[41] Furihata D. A stable and conservative finite difference scheme for the Cahn-Hilliardequation[J]. Numerische Mathematik, 2001, 87(4): 675–699.
[42] Gomez H, Hughes T J. Provably unconditionally stable, second-order timeaccurate, mixed variational methods for phase-field models[J]. Journal of Computational Physics, 2011, 230(13): 5310–5327.
[43] Guillén-González F, Tierra G. On linear schemes for a Cahn–Hilliard diffuse interface model[J]. Journal of Computational Physics, 2013, 234: 140–171.
[44] Chen L, Shen J. Applications of semi-implicit Fourier-spectral method to phasefield equations[J]. Computer Physics Communications, 1998, 108(2): 147–158.
[45] Feng X, Tang T, Yang J. Stabilized Crank-Nicolson/Adams-Bashforth schemes forphase field models[J]. East Asian Journal on Applied Mathematics, 2013, 3(1):59–80.
[46] He Y, Liu Y, Tang T. On large time-stepping methods for the Cahn–Hilliard equation[J]. Applied Numerical Mathematics, 2007, 57(5-7): 616–628.
[47] Shen J, Yang X. Numerical approximations of Allen–Cahn and Cahn–Hilliard equations[J]. Discrete and Continuous Dynamical Systems, Series A, 2010, 28(4):1669.
[48] Xu C, Tang T. Stability analysis of large time-stepping methods for epitaxial growthmodels[J]. SIAM Journal on Numerical Analysis, 2006, 44(4): 1759–1779.
[49] Cox S M, Matthews P C. Exponential time differencing for stiff systems[J]. Journalof Computational Physics, 2002, 176(2): 430–455.
[50] Hochbruck M, Ostermann A. Explicit exponential Runge–Kutta methods for semilinear parabolic problems[J]. SIAM Journal on Numerical Analysis, 2005, 43(3):1069–1090.
[51] Zhu L, Ju L, Zhao W. Fast high-order compact exponential time differencingRunge–Kutta methods for second-order semilinear parabolic equations[J]. Journal of Scientific Computing, 2016, 67: 1043–1065.
[52] Fu Z, Yang J. Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models[J]. Journal of Computational Physics, 2022, 454:110943.
[53] Tang T, Yang J. Implicit-explicit scheme for the Allen–Cahn equation preservesthe maximum principle[J]. Journal of Computational Mathematics, 2016, 34(5):471–481.
[54] Shen J, Tang T, Yang J. On the maximum principle preserving schemes for thegeneralized Allen–Cahn equation[J]. Communications in Mathematical Sciences,2016, 14(6): 1517–1534.
[55] Hou T, Tang T, Yang J. Numerical analysis of fully discretized Crank–Nicolsonscheme for fractional-in-space Allen–Cahn equations[J]. Journal of Scientific Computing, 2017, 72: 1214–1231.
[56] Du Q, Ju L, Li X, et al. Maximum principle preserving exponential time differencing schemes for the nonlocal Allen–Cahn equation[J]. SIAM Journal on numericalanalysis, 2019, 57(2): 875–898.
[57] Yang X. Linear, first and second-order, unconditionally energy stable numericalschemes for the phase field model of homopolymer blends[J]. Journal of Computational Physics, 2016, 327: 294–316.
[58] Yang X, Zhao J, Wang Q, et al. Numerical approximations for a three-componentCahn–Hilliard phase-field model based on the invariant energy quadratizationmethod[J]. Mathematical Models and Methods in Applied Sciences, 2017, 27(11):1993–2030.
[59] Cheng Q, Shen J. Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model[J]. SIAM Journal on ScientificComputing, 2018, 40(6): A3982–A4006.
[60] Shen J, Xu J, Yang J. The scalar auxiliary variable (SAV) approach for gradientflows[J]. Journal of Computational Physics, 2018, 353: 407–416.
[61] Shen J, Xu J, Yang J. A new class of efficient and robust energy stable schemes forgradient flows[J]. SIAM Review, 2019, 61(3): 474–506.
[62] Wang R, Ji Y, Shen J, et al. Application of scalar auxiliary variable scheme tophase-field equations[J]. Computational Materials Science, 2022, 212: 111556.
[63] Zhang Y, Shen J. A generalized SAV approach with relaxation for dissipative systems[J]. Journal of Computational Physics, 2022, 464: 111311.
[64] Huang F, Shen J, Wu K. Bound/positivity preserving and unconditionally stableschemes for a class of fourth order nonlinear equations[J]. Journal of ComputationalPhysics, 2022, 460: 111177.
[65] Wu K, Huang F, Shen J. A new class of higher-order decoupled schemes for theincompressible Navier-Stokes equations and applications to rotating dynamics[J].Journal of Computational Physics, 2022, 458: 111097.
[66] Cheng Q, Shen J. A new Lagrange multiplier approach for constructing structurepreserving schemes, I. Positivity preserving[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 391: 114585.
[67] Li X, Wang W, Shen J. Stability and error analysis of IMEX SAV schemes for themagneto-hydrodynamic equations[J]. SIAM Journal on Numerical Analysis, 2022,60(3): 1026–1054.
[68] Li X, Shen J, Liu Z. New SAV-pressure correction methods for the Navier-Stokesequations: stability and error analysis[J]. Mathematics of Computation, 2022,91(333): 141–167.
[69] Antoine X, Shen J, Tang Q. Scalar Auxiliary Variable/Lagrange multiplierbased pseudospectral schemes for the dynamics of nonlinear Schrödinger/GrossPitaevskii equations[J]. Journal of Computational Physics, 2021, 437: 110328.
[70] Huang F, Shen J. Bound/Positivity Preserving and Energy Stable Scalar auxiliaryVariable Schemes for Dissipative Systems: Applications to Keller–Segel and Poisson–Nernst–Planck Equations[J]. SIAM Journal on Scientific Computing, 2021,43(3): A1832–A1857.
[71] Huang F, Shen J. Stability and Error Analysis of a Class of High-Order IMEXSchemes for Navier–Stokes Equations with Periodic Boundary Conditions[J].SIAM Journal on Numerical Analysis, 2021, 59(6): 2926–2954.
[72] Li X, Shen J. On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phasefield model and its error analysis for the corresponding Cahn–Hilliard–Stokescase[J]. Mathematical Models and Methods in Applied Sciences, 2020, 30(12):2263–2297.
[73] Liu Z, Li X. The exponential scalar auxiliary variable (E-SAV) approach for phasefield models and its explicit computing[J]. SIAM Journal on Scientific Computing,2020, 42(3): B630–B655.
[74] Liu Z, Li X. Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation[J]. Numerical Algorithms, 2020, 85(1):107–132.
[75] Feng X, Tang T, Yang J. Long Time Numerical Simulations for Phase-Field Problems Using p-Adaptive Spectral Deferred Correction Methods[J]. SIAM Journalon Scientific Computing, 2015, 37(1): A271–A294.
[76] Li D, Quan C, Xu J. Stability and convergence of Strang splitting. Part I: ScalarAllen-Cahn equation[J]. Journal of Computational Physics, 2022, 458(6): 111087.
[77] Cheng Y, Kurganov A, Qu Z, et al. Fast and stable explicit operator splitting methods for phase-field models[J]. Journal of Computational Physics, 2015, 303: 45–65.
[78] Akrivis G, Li B, Li D. Energy-Decaying Extrapolated RK–SAV Methods for theAllen–Cahn and Cahn–Hilliard Equations[J]. SIAM Journal on Scientific Computing, 2019, 41(6): A3703–A3727.
[79] Huang F, Shen J, Yang Z. A Highly Efficient and Accurate New Scalar AuxiliaryVariable Approach for Gradient Flows[J]. SIAM Journal on Scientific Computing,2020, 42(4): A2514–A2536.
[80] Yang J, Yuan Z, Zhou Z. Arbitrarily High-Order Maximum Bound PreservingSchemes with Cut-off Postprocessing for Allen–Cahn Equations[J]. Journal of Scientific Computing, 2022, 90(2): 1–36.
[81] Chen H, Mao J, Shen J. Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows[J]. Numerische Mathematik, 2020,145(1): 167–196.
[82] Li X, Shen J, Rui H. Energy stability and convergence of SAV block-centeredfinite difference method for gradient flows[J]. Mathematics of Computation, 2019,88(319): 2047–2068.
[83] Pan Q, Chen C, Rabczuk T, et al. The subdivision-based IGA-EIEQ numerical scheme for the binary surfactant Cahn–Hilliard phase-field model on complexcurved surfaces[J]. Computer Methods in Applied Mechanics and Engineering,2023, 406: 115905.
[84] Pan Q, Chen C, Zhang Y J, et al. A novel hybrid IGA-EIEQ numerical method forthe Allen–Cahn/Cahn–Hilliard equations on complex curved surfaces[J]. ComputerMethods in Applied Mechanics and Engineering, 2023, 404: 115767.
[85] Yang X, He X. Numerical approximations of flow coupled binary phase field crystalsystem: Fully discrete finite element scheme with second-order temporal accuracyand decoupling structure[J]. Journal of computational physics, 2022, 467: 111448.
[86] Liu P, Ouyang Z, Chen C, et al. A novel fully-decoupled, linear, and unconditionally energy-stable scheme of the conserved Allen–Cahn phase-field model of atwo-phase incompressible flow system with variable density and viscosity[J]. Communications in Nonlinear Science and Numerical Simulation, 2022, 107: 106120.
[87] Du Q, Feng X. The phase field method for geometric moving interfaces and theirnumerical approximations[M]. Vol. 21.[S.l.]: Elsevier, 2020: 425–508.
[88] Tang T, Qiao Z. Efficient numerical methods for phase-field equations (in Chinese)[J]. Science China Mathematics, 2020, 50: 775–794.
[89] Hu Z, Wise S M, Wang C, et al. Stable and efficient finite-difference nonlinearmultigrid schemes for the phase field crystal equation[J]. Journal of ComputationalPhysics, 228(15).
[90] Wang C, Wise S M. An energy stable and convergent finite-difference scheme forthe modified phase field crystal equation[J]. SIAM Journal on Numerical Analysis,2011, 49(3): 945–969.
[91] Wise S M, Wang C, Lowengrub J S. An energy-stable and convergent finitedifference scheme for the phase field crystal equation[J]. SIAM Journal on Numerical Analysis, 2009, 47(3): 2269–2288.
[92] Du Q, Ju L, Li X, et al. Stabilized linear semi-implicit schemes for the nonlocalCahn–Hilliard equation[J]. Journal of Computational Physics, 2018, 363: 39–54.
[93] Li D, Qiao Z, Tang T. Characterizing the stabilization size for semi-implicit Fourierspectral method to phase field equations[J]. SIAM Journal on Numerical Analysis,2016, 54(3): 1653–1681.
[94] Barrett J W, Nürnberg R, Styles V. Finite element approximation of a phase fieldmodel for void electromigration[J]. SIAM Journal on Numerical Analysis, 2004,42(2): 738–772.
[95] Elliott C M, Larsson S. Error estimates with smooth and nonsmooth data for a finiteelement method for the Cahn-Hilliard equation[J]. Mathematics of Computation,1992, 58(198): 603–630.
[96] Feng X, He Y, Liu C. Analysis of finite element approximations of a phasefield model for two-phase fluids[J]. Mathematics of Computation, 2007, 76(258):539–571.
[97] Guo R, Xu Y. Local discontinuous Galerkin method and high order semi-implicitscheme for the phase field crystal equation[J]. SIAM Journal on Scientific Computing, 2016, 38(1): A105–A127.
[98] Kay D, Styles V, Süli E. Discontinuous Galerkin finite element approximation of theCahn–Hilliard equation with convection[J]. SIAM Journal on Numerical Analysis,2009, 47(4): 2660–2685.
[99] Xia Y, Xu Y, Shu C. Local discontinuous Galerkin methods for the Cahn–Hilliardtype equations[J]. Journal of Computational Physics, 2007, 227(1): 472–491.
[100] Sun Z, Wu X. A fully discrete difference scheme for a diffusion-wave system[J].Applied Numerical Mathematics, 2006, 56(2): 193–209.
[101] Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation[J]. Journal of Computational Physics, 2007, 225(2): 1533–1552.
[102] Langlands T, Henry B I. The accuracy and stability of an implicit solution methodfor the fractional diffusion equation[J]. Journal of Computational Physics, 2005,205(2): 719–736.
[103] Gao G, Sun Z, Zhang H. A new fractional numerical differentiation formula toapproximate the Caputo fractional derivative and its applications[J]. Journal ofComputational Physics, 2014, 259: 33–50.
[104] Lv C, Xu C. Error analysis of a high order method for time-fractional diffusionequations[J]. SIAM Journal on Scientific Computing, 2016, 38(5): A2699–A2724.
[105] Alikhanov A A. A new difference scheme for the time fractional diffusion equation[J]. Journal of Computational Physics, 2015, 280: 424–438.
[106] Stynes M, O’Riordan E, Gracia J L. Error analysis of a finite difference methodon graded meshes for a time-fractional diffusion equation[J]. SIAM Journal onNumerical Analysis, 2017, 55(2): 1057–1079.
[107] Kopteva N. Error analysis of the L1 method on graded and uniform meshes fora fractional-derivative problem in two and three dimensions[J]. Mathematics ofComputation, 2019, 88(319): 2135–2155.
[108] Kopteva N, Meng X. Error analysis for a fractional-derivative parabolic problemon quasi-graded meshes using barrier functions[J]. SIAM Journal on NumericalAnalysis, 2020, 58(2): 1217–1238.
[109] Liao H, McLean W, Zhang J. A Discrete Grönwall Inequality with Applicationsto Numerical Schemes for Subdiffusion Problems[J]. SIAM Journal on NumericalAnalysis, 2019, 57(1): 218–237.
[110] Kopteva N. Error analysis of an L2-type method on graded meshes for a fractionalorder parabolic problem[J]. Mathematics of Computation, 2021, 90(327): 19–40.
[111] Lubich C. Discretized fractional calculus[J]. SIAM Journal on Mathematical Analysis, 1986, 17(3): 704–719.
[112] Lubich C. Convolution quadrature and discretized operational calculus. I[J]. Numerische Mathematik, 1988, 52(2): 129–145.
[113] Lubich C. Convolution quadrature revisited[J]. BIT Numerical Mathematics, 2004,44(3): 503–514.
[114] Lubich C, Sloan I, Thomée V. Nonsmooth data error estimates for approximationsof an evolution equation with a positive-type memory term[J]. Mathematics ofComputation, 1996, 65(213): 1–17.
[115] Jin B, Li B, Zhou Z. Subdiffusion with time-dependent coefficients: improved regularity and second-order time stepping[J]. Numerische Mathematik, 2020, 145(4):883–913.
[116] Jin B, Li B, Zhou Z. Correction of high-order BDF convolution quadrature forfractional evolution equations[J]. SIAM Journal on Scientific Computing, 2017,39(6): A3129–A3152.
[117] Jin B, Li B, Zhou Z. Numerical analysis of nonlinear subdiffusion equations[J].SIAM Journal on Numerical Analysis, 2018, 56(1): 1–23.
[118] Al-Maskari M, Karaa S. Numerical approximation of semilinear subdiffusion equations with nonsmooth initial data[J]. SIAM Journal on Numerical Analysis, 2019,57(3): 1524–1544.
[119] Wang K, Zhou Z. High-order time stepping schemes for semilinear subdiffusionequations[J]. SIAM Journal on Numerical Analysis, 2020, 58(6): 3226–3250.
[120] Liao H, Tang T, Zhou T. A second-order and nonuniform time-stepping maximumprinciple preserving scheme for time-fractional Allen-Cahn equations[J]. Journalof Computational Physics, 2020, 414: 109473.
[121] Li B, Ma S. Exponential convolution quadrature for nonlinear subdiffusion equations with nonsmooth initial data[J]. SIAM Journal on Numerical Analysis, 2022,60(2): 503–528.
[122] Lubich C, Schädle A. Fast convolution for nonreflecting boundary conditions[J].SIAM Journal on Scientific Computing, 2002, 24(1): 161–182.
[123] Ke R, Ng M K, Sun H. A fast direct method for block triangular Toeplitz-likewith tri-diagonal block systems from time-fractional partial differential equations[J]. Journal of Computational Physics, 2015, 303: 203–211.
[124] Lu X, Pang H K, Sun H. Fast approximate inversion of a block triangular Toeplitzmatrix with applications to fractional sub-diffusion equations[J]. Numerical LinearAlgebra with Applications, 2015, 22(5): 866–882.
[125] Baffet D, Hesthaven J S. A kernel compression scheme for fractional differentialequations[J]. SIAM Journal on Numerical Analysis, 2017, 55(2): 496–520.
[126] Jiang S, Zhang J, Zhang Q, et al. Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations[J]. Communications inComputational Physics, 2017, 21(3): 650–678.
[127] Yan Y, Sun Z, Zhang J. Fast evaluation of the Caputo fractional derivative and itsapplications to fractional diffusion equations: a second-order scheme[J]. Communications in Computational Physics, 2017, 22(4): 1028–1048.
[128] Li X, Liao H, Zhang L. A second-order fast compact scheme with unequal timesteps for subdiffusion problems[J]. Numerical Algorithms, 2021, 86: 1011–1039.
[129] Shen J, Sun Z, Du R. Fast finite difference schemes for time-fractional diffusionequations with a weak singularity at initial time[J]. East Asian Journal on AppliedMathematics, 2018, 8(4): 834–858.
[130] Liao H, Yan Y, Zhang J. Unconditional convergence of a fast two-level linearizedalgorithm for semilinear subdiffusion equations[J]. Journal of Scientific Computing, 2019, 80(1): 1–25.
[131] Song K, Lyu P. A high-order and fast scheme with variable time steps for thetime-fractional Black-Scholes equation[J]. Mathematical Methods in the AppliedSciences, 2023, 46(2): 1990–2011.
[132] Liu N, Chen Y, Zhang J, et al. Unconditionally optimal H1-error estimate of afast nonuniform L2-1σ scheme for nonlinear subdiffusion equations[J]. NumericalAlgorithms, 92: 1655–1677.
[133] Zhu H, Xu C. A fast high order method for the time-fractional diffusion equation[J].SIAM Journal on Numerical Analysis, 2019, 57(6): 2829–2849.
[134] Karaa S. Positivity of discrete time-fractional operators with applications to phasefield equations[J]. SIAM Journal on Numerical Analysis, 2021, 59(4): 2040–2053.
[135] Quan C, Tang T, Yang J. How to Define Dissipation-Preserving Energy for TimeFractional Phase-Field Equations[J]. CSIAM Transactions on Applied Mathematics, 2020, 1(3): 478–490.
[136] Quan C, Tang T, Yang J. Numerical Energy Dissipation for Time-Fractional PhaseField Equations[J]. arXiv preprint arXiv:2009.06178, 2020.
[137] Liao H, Tang T, Zhou T. An energy stable and maximum bound preserving schemewith variable time steps for time fractional Allen–Cahn equation[J]. SIAM Journalon Scientific Computing, 2021, 43(5): A3503–A3526.
[138] Yang Y, Wang J, Chen Y, et al. Compatible L2 norm convergence of variable-stepL1 scheme for the time-fractional MBE model with slope selection[J]. Journal ofComputational Physics, 2022, 467: 111467.
[139] Arnold D N, Brezzi F, Cockburn B, et al. Unified analysis of discontinuous Galerkinmethods for elliptic problems[J]. SIAM journal on numerical analysis, 2002, 39(5):1749–1779.
[140] Brezzi F, Manzini G, Marini D, et al. Discontinuous Galerkin approximations forelliptic problems[J]. Numerical Methods for Partial Differential Equations: AnInternational Journal, 2000, 16(4): 365–378.
[141] Cockburn B, Shu C. The local discontinuous Galerkin method for time-dependentconvection-diffusion systems[J]. SIAM Journal on Numerical Analysis, 1998,35(6): 2440–2463.
[142] Ji B, Liao H, Gong Y, et al. Adaptive linear second-order energy stable schemes fortime-fractional Allen–Cahn equation with volume constraint[J]. Communicationsin Nonlinear Science and Numerical Simulation, 2020, 90: 105366.
[143] Wanner G, Hairer E. Solving ordinary differential equations II[M]. Vol. 375.[S.l.]:Springer Berlin Heidelberg New York, 1996.
[144] Hesthaven J S, Warburton T. Nodal discontinuous Galerkin methods: algorithms,analysis, and applications[M].[S.l.]: Springer Science and Business Media, 2007.
[145] Rivière B. Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation[M].[S.l.]: SIAM, 2008.
[146] Thomée V. Galerkin finite element methods for parabolic problems[M]. Vol.1054.[S.l.]: Springer, 1984.
[147] Liu C, Frank F, Rivière B. Numerical error analysis for nonsymmetric interiorpenalty discontinuous Galerkin method of Cahn–Hilliard equation[J]. NumericalMethods for Partial Differential Equations, 2019, 35(4): 1509–1537.
[148] Ciarlet P G. The finite element method for elliptic problems[M].[S.l.]: SIAM,2002.
[149] Kassam A K, Trefethen L N. Fourth-order time-stepping for stiff PDEs[J]. SIAMJournal on Scientific Computing, 2005, 26(4): 1214–1233.
[150] Chen H, Stynes M. Blow-up of error estimates in time-fractional initial-boundaryvalue problems[J]. IMA Journal of Numerical Analysis, 2021, 41(2): 974–997.
[151] Trefethen L N. Spectral methods in MATLAB[M].[S.l.]: SIAM, 2000.
[152] Shen J, Tang T, Wang L. Spectral methods: algorithms, analysis and applications[M]. Vol. 41.[S.l.]: Springer Science and Business Media, 2011.
[153] Quan C, Wu X. Global-in-Time H1-Stability of L2-1σ Method on General Nonuniform Meshes for Subdiffusion Equation[J]. Journal of Scientific Computing, 2023,95(2): 59.
[154] Quan C, Wu X, Yang J. Long time H1-stability of fast L2-1σ method ongeneral nonuniform meshes for subdiffusion equations[J]. arXiv preprint arXiv:2212.00453, 2022.