pith. sign in

arxiv: 2406.16287 · v1 · submitted 2024-06-24 · 🧮 math.NA · cs.NA

Energetic Spectral-Element Time Marching Methods for Phase-Field Nonlinear Gradient Systems

Shiqin Liu , Haijun Yu This is my paper

Pith reviewed 2026-05-23 23:41 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords spectral element methodenergetic variational GalerkinAllen-Cahn equationenergy dissipationmass conservationsuperconvergencephase-field modelstime marching
0
0 comments X

The pith

Energetic variational Galerkin spectral-element methods preserve mass conservation and energy dissipation while achieving superconvergence for phase-field systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops two spectral-element time marching schemes for nonlinear gradient systems, using the Allen-Cahn equation as the main example. The fully implicit version employs an energetic variational Galerkin form that carries over the continuous system's mass conservation and energy dissipation to the discrete level and produces superconvergence. A semi-implicit linear version uses extrapolation on the nonlinear term and recovers the same superconvergence after a few Picard iterations. Numerical tests show that the degree-three Legendre implementation outperforms both fourth-order IMEX-BDF and fourth-order ETDRK schemes on standard Allen-Cahn problems, and the methods also verify discrete mass conservation on a conservative variant of the equation.

Core claim

The central claim is that an energetic variational Galerkin formulation inside spectral-element time discretization exactly inherits the mass conservation and energy dissipation properties of the underlying continuous dynamical system, yields superconvergence, and delivers better practical performance than established fourth-order integrators when the polynomial degree is three.

What carries the argument

The energetic variational Galerkin form, which enforces discrete-level inheritance of continuous conservation and dissipation through the chosen polynomial spaces and quadrature.

If this is right

  • The methods apply directly to general large-scale nonlinear dynamical systems beyond phase-field models.
  • Discrete total mass remains conserved when the scheme is applied to conservative Allen-Cahn equations.
  • A few Picard-like iterations suffice to restore superconvergence in the semi-implicit version.
  • The approach is not limited to Allen-Cahn equations and works for other nonlinear gradient systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The preservation property could support stable long-time integration of phase-separation models without artificial drift.
  • Similar energetic variational constructions might improve structure preservation in other high-order time discretizations for gradient flows.
  • The observed advantage of degree three over fourth-order multistep and exponential integrators may depend on spatial dimension or the specific form of the nonlinearity.
  • pith_inferences

Load-bearing premise

The energetic variational Galerkin discretization exactly inherits the continuous system's conservation laws at the discrete level for the selected polynomial spaces and quadrature rules.

What would settle it

A simulation run in which the discrete energy increases over multiple time steps or total mass drifts away from its initial value would falsify the claimed inheritance of conservation properties.

Figures

Figures reproduced from arXiv: 2406.16287 by Haijun Yu, Shiqin Liu.

Figure 1
Figure 1. Figure 1: Testing ESET with Dirichlet(left) and Neumann(right) boundary condition, respectively. The scheme param [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Accuracy and efficiency of ESET, IMEX4, and ETDRK4 schemes. The common parameters used: [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The energy dissipation of ESET31 scheme ( [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effects on stability and accuracy of the stabilization constant and cut-off operation to maintain maximum [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of accuracy and efficiency between the diagonalization method and direct sparse solver of the [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solution snapshots of conservative Allen–Cahn equation ( [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mass changes (left) and energy dissipation (right) of the conservative Allen–Cahn equation and standard [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Snapshots of conservative Allen–Cahn equation for the drop coalescence test using ESET22 scheme with [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Snapshots of standard Allen–Cahn equation for the drop coalescence test using ESET22 scheme with [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mass changes (left) and energy dissipation (right) of conservative and standard (normal) Allen–Cahn equa [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen--Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen--Cahn equation, we also apply the method to a conservative Allen--Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen--Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes two energetic spectral-element time-marching methods for nonlinear gradient systems, using the Allen-Cahn equation as example. The fully implicit method employs an energetic variational Galerkin formulation claimed to preserve mass conservation and energy dissipation of the continuous system while achieving superconvergence. The semi-implicit variant uses high-order extrapolation for linearity and recovers superconvergence via Picard iterations. Numerical experiments indicate that the degree-3 Legendre version outperforms 4th-order IMEX-BDF and ETDRK schemes; the approach is also applied to a conservative Allen-Cahn equation to verify discrete mass conservation and is positioned for general nonlinear dynamical systems.

Significance. If the energetic variational Galerkin discretization rigorously inherits the continuous energy-dissipation identity and mass conservation at the algebraic level for the chosen spaces and quadrature, and if superconvergence is established, the methods would supply efficient, structure-preserving high-order integrators for phase-field models. The reported numerical outperformance and extension to conservative variants provide practical value; the variational derivation without ad-hoc parameters is a positive feature.

major comments (1)
  1. [Abstract and method formulation] Abstract and method formulation: the central claim that the energetic variational Galerkin form 'can maintain the mass conservation and energy dissipation property of the continuous dynamical system' is load-bearing for the paper's contribution, yet the manuscript must supply the explicit algebraic steps demonstrating that the discrete inner product, test-function space, and quadrature exactly reproduce the integration-by-parts identity and nonlinear potential without remainder; for the cubic nonlinearity of Allen-Cahn this is not automatic and requires verification that no quadrature error appears in the energy law.
minor comments (2)
  1. [Numerical experiments] Numerical experiments section: the claim that the degree-3 version 'outperforms' the 4th-order IMEX-BDF and ETDRK methods should include tabulated error values, observed convergence rates, and precise problem parameters (mesh size, time-step range, tolerance) to allow direct comparison.
  2. [Abstract] Abstract: the term 'superconvergence' is invoked without stating the observed temporal order or referencing the supporting analysis or tables.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment on providing explicit algebraic verification of the discrete energy law is addressed below.

read point-by-point responses

  1. Referee: [Abstract and method formulation] Abstract and method formulation: the central claim that the energetic variational Galerkin form 'can maintain the mass conservation and energy dissipation property of the continuous dynamical system' is load-bearing for the paper's contribution, yet the manuscript must supply the explicit algebraic steps demonstrating that the discrete inner product, test-function space, and quadrature exactly reproduce the integration-by-parts identity and nonlinear potential without remainder; for the cubic nonlinearity of Allen-Cahn this is not automatic and requires verification that no quadrature error appears in the energy law.

    Authors: We agree that an explicit algebraic verification is required to substantiate the central claim, particularly for the cubic nonlinearity where quadrature effects are not automatic. The current manuscript presents the energetic variational formulation and states the preservation properties but does not include the full step-by-step algebraic expansion showing exact reproduction of the integration-by-parts identity and nonlinear potential without remainder. In the revised version we will insert a dedicated subsection that carries out this derivation for the chosen spectral-element spaces and quadrature rules, confirming that the discrete energy dissipation law holds exactly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on independent variational construction and external numerical benchmarks.

full rationale

The paper constructs the energetic variational Galerkin form from first-principles variational principles applied to the continuous gradient system, then discretizes in a spectral-element time basis. The claimed discrete mass conservation and energy dissipation follow directly from the algebraic structure of the chosen test spaces and quadrature (stated as an exact inheritance), without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. Performance claims are supported by direct comparison to independent fourth-order IMEX-BDF and ETDRK schemes on Allen-Cahn problems, which are external benchmarks. No step in the provided derivation chain collapses to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of Galerkin methods and polynomial approximation theory plus the unverified assertion that the energetic variational form exactly preserves the continuous invariants. No free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption The energetic variational Galerkin form inherits the continuous system's mass conservation and energy dissipation exactly at the discrete level.
    This is presented as a key advantage of the implicit method; its justification is not detailed in the abstract.
  • standard math Standard spectral-element approximation theory and quadrature rules apply without additional consistency errors that would violate conservation.
    Implicit in any spectral-element discretization.

pith-pipeline@v0.9.0 · 5772 in / 1445 out tokens · 17636 ms · 2026-05-23T23:41:40.221993+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (2) (1958) 258–267. doi:10.1063/1.1744102

    work page doi:10.1063/1.1744102 1958

  2. [2]

    S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its appli- cation to antiphase domain coarsening, Acta Met. Mater 27 (1979) 1085–1095. doi:10.1016/ 0001-6160(79)90196-2

    work page 1979

  3. [3]

    Chen, Phase-field models for microstructure evolution, Annu

    L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32 (1) (2002) 113–140. doi:10.1146/annurev.matsci.32.112001.132041

    work page doi:10.1146/annurev.matsci.32.112001.132041 2002

  4. [4]

    Greenwood, N

    M. Greenwood, N. Provatas, J. Rottler, Free energy functionals for efficient phase field crystal modeling of structural phase transformations, Phys. Rev. Lett. 105 (4) (2010). doi:10.1103/ PhysRevLett.105.045702

    work page 2010

  5. [5]

    D. M. Anderson, G. B. McFadden, A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30 (1) (1998) 139–165. doi:10.1146/annurev.fluid.30.1.139

    work page doi:10.1146/annurev.fluid.30.1.139 1998

  6. [6]

    Lowengrub, L

    J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological tran- sitions, Proceedings: Mathematical, Physical and Engineering Sciences 454 (1978) (1998) 2617–

    work page 1978

  7. [7]

    arXiv:53238, doi:10.1098/rspa.1998.0273. 24

    work page doi:10.1098/rspa.1998.0273 1998

  8. [8]

    P. Yue, J. J. Feng, C. Liu, J. Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech. 515 (2004) 293–317. doi:10.1017/S0022112004000370

    work page doi:10.1017/s0022112004000370 2004

  9. [9]

    Qian, X.-P

    T. Qian, X.-P. Wang, P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech. 564 (2006) 333–360. doi:10.1017/S0022112006001935

    work page doi:10.1017/s0022112006001935 2006

  10. [11]

    J. T. Oden, A. Hawkins, S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci. 20 (03) (2010) 477–517. doi:10.1142/S0218202510004313

    work page doi:10.1142/s0218202510004313 2010

  11. [12]

    S. M. Wise, J. S. Lowengrub, H. B. Frieboes, V. Cristini, Three-dimensional multispecies non- linear tumor growth I: Model and numerical method, J. Theor. Biol. 253 (3) (2008) 524–543. doi:10.1016/j.jtbi.2008.03.027

    work page doi:10.1016/j.jtbi.2008.03.027 2008

  12. [13]

    Miranville, E

    A. Miranville, E. Rocca, G. Schimperna, On the long time behavior of a tumor growth model, J. Differ. Equations 267 (4) (2019) 2616–2642. doi:10.1016/j.jde.2019.03.028

    work page doi:10.1016/j.jde.2019.03.028 2019

  13. [14]

    C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D 179 (3-4) (2003) 211–228. doi: 10.1016/S0167-2789(03)00030-7

    work page doi:10.1016/s0167-2789(03)00030-7 2003

  14. [15]

    Q. Du, R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28 (5) (1991) 1310–1322. doi:10.1137/0728069

    work page doi:10.1137/0728069 1991

  15. [16]

    C. M. Elliott, A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30 (6) (1993) 1622–1663. doi:10.1137/0730084

    work page doi:10.1137/0730084 1993

  16. [17]

    D. J. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation, MRS Proc. 529 (1998) 39. doi:10.1557/PROC-529-39

    work page doi:10.1557/proc-529-39 1998

  17. [18]

    C. Wang, S. M. Wise, An energy stable and convergent finite-difference scheme for the modi- fied phase field crystal equation, SIAM J. Numer. Anal. 49 (3) (2011) 945–969. doi:10.1137/ 090752675

    work page 2011

  18. [19]

    J. Shen, C. Wang, X. Wang, S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich–Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal. 50 (1) (2012) 105–125. doi:10.1137/110822839

    work page doi:10.1137/110822839 2012

  19. [20]

    L. Chen, J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun. 108 (2-3) (1998) 147–158. doi:10.1016/S0010-4655(97)00115-X

    work page doi:10.1016/s0010-4655(97)00115-x 1998

  20. [21]

    Zhu, L.-Q

    J. Zhu, L.-Q. Chen, J. Shen, V. Tikare, Coarsening kinetics from a variable-mobility Cahn– Hilliard equation: Application of a semi-implicit Fourier spectral method, Phys. Rev. E 60 (4) (1999) 3564–3572. doi:10.1103/PhysRevE.60.3564

    work page doi:10.1103/physreve.60.3564 1999

  21. [22]

    C. Xu, T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal. 44 (4) (2006) 1759–1779. doi:10.1137/050628143

    work page doi:10.1137/050628143 2006

  22. [23]

    J. Shen, X. Yang, Numerical approximations of Allen–Cahn and Cahn–Hilliard equations, Dis- crete Cont. Dyn. A 28 (4) (2010) 1669–1691. doi:10.3934/dcds.2010.28.1669

    work page doi:10.3934/dcds.2010.28.1669 2010

  23. [24]

    L. Wang, H. Yu, On efficient second order stabilized semi-implicit schemes for the Cahn– Hilliard phase-field equation, J. Sci. Comput. 77 (2) (2018) 1185–1209. doi:10.1007/ s10915-018-0746-2 . 25

    work page 2018

  24. [25]

    L. Wang, H. Yu, An energy stable linear diffusive Crank–Nicolson scheme for the Cahn–Hilliard gradient flow, J. Comput. Appl. Math. 377 (2020) 112880. doi:10.1016/j.cam.2020.112880

    work page doi:10.1016/j.cam.2020.112880 2020

  25. [26]

    X. Li, Z. Qiao, C. Wang, Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn–Hilliard equation, Sci. China Math. (2023). doi:10. 1007/s11425-022-2036-8

    work page 2023

  26. [27]

    R. Guo, F. Filbet, Y. Xu, Efficient high order semi-implicit time discretization and local discon- tinuous Galerkin methods for highly nonlinear PDEs, J. Sci. Comput. 68 (3) (2016) 1029–1054. doi:10.1007/s10915-016-0170-4

    work page doi:10.1007/s10915-016-0170-4 2016

  27. [28]

    X. Wang, L. Ju, Q. Du, Efficient and stable exponential time differencing Runge–Kutta methods for phase field elastic bending energy models, J. Comput. Phys. 316 (2016) 21–38. doi:10.1016/ j.jcp.2016.04.004

    work page 2016

  28. [29]

    L. Zhu, L. Ju, W. Zhao, Fast high-order compact exponential time differencing Runge–Kutta methods for second-order semilinear parabolic equations, J. Sci. Comput. 67 (3) (2016) 1043–

    work page 2016

  29. [30]

    doi:10.1007/s10915-015-0117-1

    work page doi:10.1007/s10915-015-0117-1

  30. [32]

    Zhang, J

    H. Zhang, J. Yan, X. Qian, X. Chen, S. Song, Explicit third-order unconditionally structure- preserving schemes for conservative Allen–Cahn equations, J. Sci. Comput. 90 (1) (2021) 8. doi:10.1007/s10915-021-01691-w

    work page doi:10.1007/s10915-021-01691-w 2021

  31. [33]

    L. Ju, X. Li, Z. Qiao, J. Yang, Maximum bound principle preserving integrating factor Runge– Kutta methods for semilinear parabolic equations, J. Comput. Phys. 439 (2021) 110405. arXiv: 2010.12165, doi:10.1016/j.jcp.2021.110405

    work page doi:10.1016/j.jcp.2021.110405 2021

  32. [34]

    Z. Fu, T. Tang, J. Yang, Energy plus maximum bound preserving Runge–Kutta methods for the Allen–Cahn equation, J. Sci. Comput. 92 (3) (2022) 97. arXiv:2203.04784, doi:10.1007/ s10915-022-01940-6

    work page arXiv 2022

  33. [35]

    X. Yang, J. Zhao, X. He, Linear, second order and unconditionally energy stable schemes for the viscous Cahn–Hilliard equation with hyperbolic relaxation using the invariant energy quadrati- zation method, J. Comput. Appl. Math. 343 (2018) 80–97. doi:10.1016/j.cam.2018.04.027

    work page doi:10.1016/j.cam.2018.04.027 2018

  34. [36]

    X. Yang, H. Yu, Efficient second order unconditionally stable schemes for a phase field moving contact line model using an invariant energy quadratization approach, SIAM J. Sci. Comput. 40 (3) (2018) B889–B914. doi:10.1137/17M1125005

    work page doi:10.1137/17m1125005 2018

  35. [37]

    Yang, G.-D

    X. Yang, G.-D. Zhang, Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn–Hilliard and Allen–Cahn equations with general nonlinear po- tential, J. Sci. Comput. 82 (3) (2020) 55. doi:10.1007/s10915-020-01151-x

    work page doi:10.1007/s10915-020-01151-x 2020

  36. [38]

    Guill´ en-Gonz´ alez, G

    F. Guill´ en-Gonz´ alez, G. Tierra, On linear schemes for a Cahn–Hilliard diffuse interface model, J. Comput. Phys. 234 (2013) 140–171. doi:10.1016/j.jcp.2012.09.020

    work page doi:10.1016/j.jcp.2012.09.020 2013

  37. [39]

    Guill´ en-Gonz´ alez, G

    F. Guill´ en-Gonz´ alez, G. Tierra, Second order schemes and time-step adaptivity for Allen–Cahn and Cahn–Hilliard models, Comput. Math. Appl. 68 (8) (2014) 821–846. doi:10.1016/j.camwa. 2014.07.014

    work page doi:10.1016/j.camwa 2014

  38. [40]

    J. Shen, J. Xu, J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev. 61 (3) (2019) 474–506. doi:10.1137/17M1150153. 26

    work page doi:10.1137/17m1150153 2019

  39. [41]

    J. Shen, J. Xu, J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys. 353 (2018) 407–416. doi:10.1016/j.jcp.2017.10.021

    work page doi:10.1016/j.jcp.2017.10.021 2018

  40. [42]

    J. Shen, J. Xu, Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows, SIAM J. Numer. Anal. 56 (5) (2018) 2895–2912. doi:10.1137/17M1159968

    work page doi:10.1137/17m1159968 2018

  41. [43]

    X. Feng, A. Prohl, Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows, Numer. Math. 94 (1) (2003) 33–65. doi:10.1007/s00211-002-0413-1

    work page doi:10.1007/s00211-002-0413-1 2003

  42. [44]

    L. Wang, H. Yu, Energy-stable second-order linear schemes for the Allen–Cahn phase-field equa- tion, Commun. Math. Sci. 17 (3) (2019) 609–635. doi:20190830152653

    work page 2019

  43. [45]

    Tang, H.-p

    J.-g. Tang, H.-p. Ma, Single and multi-interval Legendre spectral methods in time for parabolic equations, Numer. Meth. Part. D. E. 22 (5) (2006) 1007–1034. doi:10.1002/num.20135

    work page doi:10.1002/num.20135 2006

  44. [46]

    Shen, L.-L

    J. Shen, L.-L. Wang, Fourierization of the Legendre–Galerkin method and a new space-time spectral method, Appl. Numer. Math. 57 (5) (2007) 710–720. doi:10.1016/j.apnum.2006.07. 012

    work page doi:10.1016/j.apnum.2006.07 2007

  45. [47]

    Kessler, R

    D. Kessler, R. H. Nochetto, A. Schmidt, A posteriori error control for the Allen–Cahn problem: Circumventing Gronwall’s inequality, ESAIM Math. Model. Numer. Anal. 38 (01) (2004) 129–142. doi:10.1051/m2an:2004006

    work page doi:10.1051/m2an:2004006 2004

  46. [48]

    Condette, C

    N. Condette, C. Melcher, E. S¨ uli, Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth, Math. Comp. 80 (273) (2011) 205–223. doi:10.1090/S0025-5718-10-02365-3

    work page doi:10.1090/s0025-5718-10-02365-3 2011

  47. [49]

    Shen, C.-T

    J. Shen, C.-T. Sheng, An efficient space-time method for time fractional diffusion equation, J. Sci. Comput. 81 (2) (2019) 1088–1110. doi:10.1007/s10915-019-01052-8

    work page doi:10.1007/s10915-019-01052-8 2019

  48. [50]

    Shen, Efficient spectral-Galerkin method I

    J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15 (6) (1994) 1489. doi:10.1137/0915089

    work page doi:10.1137/0915089 1994

  49. [51]

    H. Yu, X. Yang, Numerical approximations for a phase-field moving contact line model with variable densities and viscosities, J. Comput. Phys. 334 (2017) 665–686. doi:10.1016/j.jcp. 2017.01.026

    work page doi:10.1016/j.jcp 2017

  50. [52]

    L. Wang, H. Yu, Convergence analysis of an unconditionally energy stable linear Crank–Nicolson scheme for the Cahn–Hilliard equation, J. Math. Study 51 (1) (2018) 89–114. doi:10.4208/jms. v51n1.18.06

    work page doi:10.4208/jms 2018

  51. [53]

    Tang, H.-p

    J.-g. Tang, H.-p. Ma, Single and multi-interval legendreτ-methods in time for parabolic equations, Adv. Comput. Math. 17 (4) (2002) 349–367. doi:10.1023/A:1016273820035

    work page doi:10.1023/a:1016273820035 2002

  52. [54]

    R. E. Lynch, J. R. Rice, D. H. Thomas, Direct solution of partial difference equations by tensor product methods, Numer. Math. 6 (1) (1964) 185–199. doi:10.1007/BF01386067

    work page doi:10.1007/bf01386067 1964

  53. [55]

    D. Kong, J. Shen, L.-L. Wang, S. Xiang, Eigenvalue analysis and applications of the Legendre dual-Petrov–Galerkin methods for initial value problems, arXiv:2211.10599 (2022)

    work page arXiv 2022

  54. [56]

    Z. Chen, Y. Liu, Efficient and parallel solution of high-order continuous time Galerkin for dis- sipative and wave propagation problems, arXiv:2303.05008 (2023). doi:10.48550/arXiv.2303. 05008

    work page doi:10.48550/arxiv.2303 2023

  55. [57]

    J. Shen, T. Tang, L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, no. 41 in Springer Series in Computational Mathematics, Springer, Heidelberg ; New York, 2011. 27

    work page 2011

  56. [58]

    Canuto, M

    C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer Science & Business Media, 2007

    work page 2007

  57. [59]

    Dupont, A unified theory of superconvergence for Galerkin methods for two-point boundary problems, SIAM J

    T. Dupont, A unified theory of superconvergence for Galerkin methods for two-point boundary problems, SIAM J. Numer. Anal. 13 (3) (1976) 362–368. doi:10.1137/0713032

    work page doi:10.1137/0713032 1976

  58. [60]

    D. N. Arnold, J. Douglas, Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable, Calcolo 16 (4) (1979) 345–369. doi:10.1007/ BF02576636

    work page 1979

  59. [61]

    Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer

    T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math. 61 (1) (1992) 373–382. doi:10.1007/BF01385515

    work page doi:10.1007/bf01385515 1992

  60. [62]

    A. Zhou, Q. Lin, Optimal and superconvergence estimates of the finite element method for a scalar hyperbolic equation, Acta Math. Sci. 14 (1) (1994) 90–94. doi:10.1016/S0252-9602(18) 30094-8

    work page doi:10.1016/s0252-9602(18 1994

  61. [63]

    Karakashian, C

    O. Karakashian, C. Makridakis, A space-time finite element method for the nonlinear Schr¨ odinger equation: The continuous Galerkin method, SIAM J. Numer. Anal. 36 (6) (1999) 1779–1807. doi:10.1137/S0036142997330111

    work page doi:10.1137/s0036142997330111 1999

  62. [64]

    Zhang, Superconvergence of spectral collocation and p-version methods in one dimensional problems, Math

    Z. Zhang, Superconvergence of spectral collocation and p-version methods in one dimensional problems, Math. Comp. 74 (252) (2005) 1621–1636. doi:10.1090/S0025-5718-05-01756-4

    work page doi:10.1090/s0025-5718-05-01756-4 2005

  63. [65]

    Huang, H

    Q. Huang, H. Xie, H. Brunner, Superconvergence of discontinuous Galerkin solutions for delay differential equations of pantograph type, SIAM J. Sci. Comput. 33 (5) (2011) 2664–2684. doi: 10.1137/110824632

    work page doi:10.1137/110824632 2011

  64. [66]

    Q. Gu, Y. Chen, Y. Huang, Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations, Comp. Appl. Math. 41 (8) (2022) 361. doi:10. 1007/s40314-022-02070-3

    work page 2022

  65. [67]

    Zhang, L

    M. Zhang, L. Yi, Superconvergent postprocessing of the continuous Galerkin time stepping method for nonlinear initial value problems with application to parabolic problems, J. Sci. Com- put. 94 (2) (2023) 31. doi:10.1007/s10915-022-02086-1

    work page doi:10.1007/s10915-022-02086-1 2023

  66. [68]

    Kassam, Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (4) (2005) 1214–1233. doi:10.1137/s1064827502410633

    work page doi:10.1137/s1064827502410633 2005

  67. [69]

    B. Li, J. Yang, Z. Zhou, Arbitrarily high-order exponential cut-off methods for preserving max- imum principle of parabolic equations, SIAM J. Sci. Comput. 42 (6) (2020) A3957–A3978. doi:10.1137/20M1333456

    work page doi:10.1137/20m1333456 2020

  68. [70]

    J. Yang, Z. Yuan, Z. Zhou, Arbitrarily high-order maximum bound preserving schemes with cut- off postprocessing for Allen–Cahn equations, J. Sci. Comput. 90 (2) (2022) 76. doi:10.1007/ s10915-021-01746-y

    work page 2022

  69. [71]

    Rubinstein, P

    J. Rubinstein, P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (3) (1992) 249–264. doi:10.1093/imamat/48.3.249. 28

    work page doi:10.1093/imamat/48.3.249 1992