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arxiv: 2604.18945 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

Relaxed Generalized Scalar Auxiliary Variable Exponential Integrator for A Modified Landau-de Gennes Theory for Smectic Liquid Crystals

Wenshuai Hu , Guanghua Ji , Xiao Li This is my paper

Pith reviewed 2026-05-10 02:35 UTC · model grok-4.3

classification 🧮 math.NA cs.NA MSC 65M1265M1576A15
keywords smectic liquid crystalsmodified Landau-de Gennes modelenergy stable schemeexponential integratorscalar auxiliary variablenumerical analysistopological defectsunconditional stability
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The pith

A relaxed generalized scalar auxiliary variable exponential integrator produces unconditionally energy-stable and optimally accurate simulations of the modified Landau-de Gennes model for smectic liquid crystals by removing restrictive CFL,

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

The paper introduces a numerical scheme for the coupled system that models the Smectic-A phase, where a tensor order parameter Q tracks molecular orientation and a scalar u tracks positional layering. The method merges the generalized scalar auxiliary variable approach with exponential integrators and adds a relaxed correction that rewrites the time discretization as an equivalent quasi-implicit backward Euler form. This change removes the mesh-ratio restrictions that previously limited practical use of the original GSAV-EI method while preserving its stability features. The authors prove that the resulting scheme remains unconditionally stable with respect to a modified discrete energy, keeps the tensor solutions uniformly bounded, and attains optimal convergence rates in both time and space. These properties matter for long-time simulations that track the formation and motion of topological defects without artificial step-size limits.

Core claim

The authors establish that the relaxed GSAV-EI scheme for the coupled mLdG system is unconditionally energy stable with respect to a modified discrete energy, that the numerical solutions for the tensor Q remain uniformly bounded, and that it achieves optimal error estimates in time and space. This follows from reformulating the exponential time differencing into an equivalent quasi-implicit backward Euler-type structure that eliminates CFL restrictions without introducing additional errors or losing the original stability properties of the GSAV-EI method.

What carries the argument

The relaxed generalized scalar auxiliary variable exponential integrator (GSAV-EI), obtained by rewriting exponential time differencing as a quasi-implicit backward Euler structure that carries the unconditional stability and error analysis.

If this is right

  • The numerical solutions for the tensor order parameter Q remain uniformly bounded for all time steps.
  • The scheme preserves a modified discrete energy unconditionally, independent of the mesh ratio.
  • Optimal error estimates hold simultaneously in time and space for the fully discrete solutions.
  • The method can simulate complex topological defect dynamics in the smectic phase over long times without artificial step-size restrictions.

Where Pith is reading between the lines

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

  • The same reformulation strategy could be tested on other exponential-integrator-based schemes for nonlinear gradient flows that currently require CFL conditions.
  • Uniform boundedness of Q may allow the method to run for arbitrarily long times while remaining physically meaningful, a property useful for studying slow coarsening or defect annihilation.
  • The approach might extend directly to three-dimensional versions of the mLdG model or to related Landau-de Gennes systems with additional couplings.

Load-bearing premise

The reformulation of the exponential time differencing into an equivalent quasi-implicit backward Euler-type structure eliminates the restrictive CFL mesh-ratio conditions without introducing additional errors or losing the original stability properties of the GSAV-EI method.

What would settle it

A computation on a sequence of successively refined meshes using time steps that violate the original CFL bound, where either the discrete energy increases or the observed convergence rates fall below first-order in time and second-order in space, would falsify the unconditional stability and optimal error claims.

read the original abstract

The Smectic-A (SmA) phase is modeled by a modified Landau-de Gennes (mLdG) model proposed by Xia et al. [Phys. Rev. Lett., 126 (2021), 177801], in which a tensor order parameter Q for the orientational order is coupled with a real scalar $u$ characterizing the positional order. In this paper, we propose and analyze a novel, highly efficient, and unconditionally energy-stable numerical scheme for this coupled system by combining the generalized scalar auxiliary variable-exponential integrator (GSAV-EI) approach with a relaxed correction strategy. In particular, we reformulate the exponential time differencing time discretization into an equivalent quasi-implicit backward Euler-type structure, a pivotal step that eliminates the restrictive CFL mesh-ratio conditions of the original GSAV-EI method and enables a rigorous fully discrete error analysis. Theoretically, we rigorously establish the unconditional energy stability with respect to a modified discrete energy and the uniform boundedness of the numerical solutions Q, along with optimal error estimates in both time and space. Comprehensive numerical experiments are presented to demonstrate the accuracy, efficiency, and structural preservation of the algorithm, as well as its capability in capturing complex topological defect dynamics.

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

2 major / 2 minor

Summary. The paper proposes a relaxed GSAV-EI scheme for the coupled mLdG system modeling smectic-A liquid crystals, where tensor Q is coupled to scalar u. The key step is reformulating the exponential time differencing into an equivalent quasi-implicit backward Euler structure to remove CFL restrictions, followed by proofs of unconditional energy stability w.r.t. a modified discrete energy, uniform boundedness of Q, and optimal error estimates in time and space, with supporting numerical experiments on accuracy and defect dynamics.

Significance. If the reformulation equivalence holds exactly and the fully discrete analysis is rigorous, the work would offer a practical, unconditionally stable integrator for long-time simulations of nonlinear coupled systems in liquid crystal modeling, removing mesh-ratio constraints while preserving structure and achieving optimal convergence. This addresses a common bottleneck in exponential integrators for such models.

major comments (2)
  1. [Reformulation and equivalence claim] The reformulation step (abstract and the time-discretization section): the assertion that rewriting the GSAV-EI exponential differencing as an equivalent quasi-implicit backward Euler form eliminates CFL conditions without introducing additional errors or altering stability for the nonlinear Q-u coupling is load-bearing. Explicit verification is required that the rewritten scheme is identical to the original on the coupled operators, as any discrepancy would prevent direct transfer of the unconditional stability and error estimates.
  2. [Energy stability proof] Stability analysis section: unconditional energy stability is shown only w.r.t. a modified discrete energy. The relation between this modified energy and the original continuous energy functional must be clarified to confirm that the stability result implies the desired physical properties without artificial dissipation.
minor comments (2)
  1. [Abstract] Abstract: the order of the optimal error estimates (e.g., O(Δt) in time and O(h^2) in space) should be stated explicitly for clarity.
  2. [Numerical results] Numerical experiments: include tabulated convergence rates and direct comparisons to the original GSAV-EI or other schemes to quantify the efficiency gain from the relaxed correction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Reformulation and equivalence claim] The reformulation step (abstract and the time-discretization section): the assertion that rewriting the GSAV-EI exponential differencing as an equivalent quasi-implicit backward Euler form eliminates CFL conditions without introducing additional errors or altering stability for the nonlinear Q-u coupling is load-bearing. Explicit verification is required that the rewritten scheme is identical to the original on the coupled operators, as any discrepancy would prevent direct transfer of the unconditional stability and error estimates.

    Authors: We thank the referee for emphasizing the importance of verifying the equivalence. The reformulation in Section 3 is obtained by exact algebraic rearrangement of the exponential integrator terms applied to the linear operator, yielding an equivalent quasi-implicit backward Euler structure for the coupled system. In the revised manuscript we will insert a new lemma immediately after the scheme definition that explicitly demonstrates the two formulations produce identical updates for both Q and u at each time step. Because the equivalence is algebraic and holds without approximation, the unconditional stability and optimal error estimates transfer directly with no additional truncation error introduced by the rewriting. revision: yes

  2. Referee: [Energy stability proof] Stability analysis section: unconditional energy stability is shown only w.r.t. a modified discrete energy. The relation between this modified energy and the original continuous energy functional must be clarified to confirm that the stability result implies the desired physical properties without artificial dissipation.

    Authors: We appreciate the referee's request for clarification on the energy relation. The modified discrete energy includes the non-negative SAV relaxation term, which is controlled uniformly by the relaxation parameter. In the revised version we will add a remark following Theorem 3.1 that establishes the equivalence (up to constants independent of the time step) between the modified discrete energy and the original discrete energy functional. We will further note that the dissipation inequality for the modified energy implies dissipation of the original energy up to a term that vanishes as the relaxation parameter tends to zero, thereby confirming that the scheme respects the physical energy law without introducing persistent artificial dissipation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper proposes a reformulation of the GSAV-EI time discretization into an equivalent quasi-implicit backward Euler structure for the coupled mLdG system, then proves unconditional energy stability with respect to a modified discrete energy, uniform boundedness of Q, and optimal error estimates. These proofs rely on standard techniques in numerical analysis for energy-stable schemes and do not reduce by construction to the inputs or to self-citations. The model originates from an external citation (Xia et al.), the GSAV-EI base is combined with a new relaxed correction, and the equivalence claim enables but does not presuppose the stability result. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citation chains are present; the derivation is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters or assumptions; the scheme rests on the external mLdG model and standard numerical analysis assumptions such as solution regularity for error bounds.

axioms (1)
  • domain assumption The modified Landau-de Gennes model proposed by Xia et al. accurately represents the SmA phase.
    The numerical scheme is constructed directly on this model without re-deriving its validity.

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Reference graph

Works this paper leans on

161 extracted references · 161 canonical work pages

  1. [1]

    Zhao, Jia and Wang, Qi , title =. J. Sci. Comput. , month = sep, pages =. 2016 , issue_date =. doi:10.1007/s10915-016-0177-x , abstract =

    work page doi:10.1007/s10915-016-0177-x 2016

  2. [2]

    Physical review letters , volume=

    Structural landscapes in geometrically frustrated smectics , author=. Physical review letters , volume=. 2021 , publisher=

    work page 2021

  3. [3]

    ESAIM: Mathematical Modelling and Numerical Analysis , volume=

    Variational and numerical analysis of a Q-tensor model for smectic-A liquid crystals , author=. ESAIM: Mathematical Modelling and Numerical Analysis , volume=. 2023 , publisher=

    work page 2023

  4. [4]

    SIAM Journal on Applied Mathematics , volume=

    A modified Landau--de Gennes theory for smectic liquid crystals: phase transitions and structural transitions , author=. SIAM Journal on Applied Mathematics , volume=. 2025 , publisher=

    work page 2025

  5. [5]

    , title =

    Ji, G. , title =. East Asian Journal on Applied Mathematics , year =

    work page

  6. [6]

    and Zhu, X

    Mo, M. and Zhu, X. , title =. Mathematics of Computation , year =

    work page

  7. [7]

    doi:10.1098/rspa.2023.0813 , adsurl =

    Proceedings of the Royal Society of London Series A , year = 2024, month = jun, volume =. doi:10.1098/rspa.2023.0813 , adsurl =

    work page doi:10.1098/rspa.2023.0813 2024

  8. [8]

    and Atherton, T

    Adler, J. and Atherton, T. and Emerson, D. and MacLachlan, S. , title =. SIAM J. Numer. Anal. , volume =

    work page

  9. [9]

    SIAM Journal on Scientific Computing , volume =

    Energy-decaying extrapolated RK--SAV methods for the Allen--Cahn and Cahn--Hilliard equations , author =. SIAM Journal on Scientific Computing , volume =. 2019 , publisher =

    work page 2019

  10. [10]

    Liquid Crystals , volume =

    Numerical results for the blue phases , author =. Liquid Crystals , volume =. 2009 , publisher =

    work page 2009

  11. [11]

    , title =

    Alouges, F. , title =. SIAM J. Numer. Anal. , volume =

    work page

  12. [12]

    Gradient Flows: In Metric Spaces and in the Space of Probability Measures , year =

    Ambrosio, Luigi and Gigli, Nicola and Savar. Gradient Flows: In Metric Spaces and in the Space of Probability Measures , year =

    work page

  13. [13]

    2001 , publisher =

    Chirality in liquid crystals , author =. 2001 , publisher =

    work page 2001

  14. [14]

    Journal of computational physics , volume =

    A new class of time discretization schemes for the solution of nonlinear PDEs , author =. Journal of computational physics , volume =. 1998 , publisher =

    work page 1998

  15. [15]

    Soviet Physics Uspekhi , abstract =

    V A Belyakov and Vladimir E Dmitrienko , title =. Soviet Physics Uspekhi , abstract =. 1985 , month =. doi:10.1070/PU1985v028n07ABEH003870 , url =

    work page doi:10.1070/pu1985v028n07abeh003870 1985

  16. [16]

    and Wood, R

    Bonet, J. and Wood, R. D. , title =

    work page

  17. [17]

    and Shen, J

    Cai, Y. and Shen, J. and Xu, X. , title =. Math. Models Methods Appl. Sci. , volume =

    work page

  18. [18]

    and Majumdar, A

    Canevari, G. and Majumdar, A. and Spicer, A. , title =. SIAM J. Appl. Math. , volume =

    work page

  19. [19]

    Applied Physics Letters , volume =

    Electric field-induced monodomain blue phase liquid crystals , author =. Applied Physics Letters , volume =. 2013 , publisher =

    work page 2013

  20. [20]

    and Hardt, R

    Cohen, R. and Hardt, R. and Kinderlehrer, D. and Lin, S. and Luskin, M. , title =. Theory and Applications of Liquid Crystals , pages =

    work page

  21. [21]

    Journal of Computational Physics , volume =

    Exponential time differencing for stiff systems , author =. Journal of Computational Physics , volume =. 2002 , publisher =

    work page 2002

  22. [22]

    1974 , publisher =

    de Gennes, Pierre-Gilles , title =. 1974 , publisher =

    work page 1974

  23. [23]

    BIT Numerical Mathematics , year =

    Low regularity integrators for semilinear parabolic equations with maximum bound principles , author =. BIT Numerical Mathematics , year =

    work page

  24. [24]

    SIAM Journal on Numerical Analysis , volume =

    Huang, Fukeng and Shen, Jie , title =. SIAM Journal on Numerical Analysis , volume =. 2024 , doi =. https://doi.org/10.1137/23M1612986 , abstract =

    work page doi:10.1137/23m1612986 2024

  25. [25]

    Journal of Computational Physics , volume =

    Stabilized linear semi-implicit schemes for the nonlocal Cahn--Hilliard equation , author =. Journal of Computational Physics , volume =. 2018 , publisher =

    work page 2018

  26. [26]

    SIAM Journal on Numerical Analysis , volume =

    Du, Qiang and Ju, Lili and Li, Xiao and Qiao, Zhonghua , title =. SIAM Journal on Numerical Analysis , volume =. 2019 , doi =. https://doi.org/10.1137/18M118236X , abstract =

    work page doi:10.1137/18m118236x 2019

  27. [27]

    2021 , doi =

    Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes , journal =. 2021 , doi =

    work page 2021

  28. [28]

    Physical Review E , volume =

    Numerical calculations of the phase diagram of cubic blue phases in cholesteric liquid crystals , author =. Physical Review E , volume =. 2005 , publisher =

    work page 2005

  29. [29]

    Mathematics of Computation , year =

    Feng, Yue and Maierhofer, Georg and Schratz, Katharina , title =. Mathematics of Computation , year =. doi:10.1090/mcom/3922 , note =

    work page doi:10.1090/mcom/3922

  30. [30]

    Numerische Mathematik , year =

    Feng, Xiaobing and Prohl, Andreas , title =. Numerische Mathematik , year =

    work page

  31. [31]

    East Asian Journal on Applied Mathematics , year =

    Feng, Xiaobing and Tang, Tao and Yang, Jiang , title =. East Asian Journal on Applied Mathematics , year =

    work page

  32. [32]

    SIAM Journal on Mathematical Analysis , volume =

    Weak time regularity and uniqueness for a Q-tensor model , author =. SIAM Journal on Mathematical Analysis , volume =. 2014 , publisher =

    work page 2014

  33. [33]

    Liquid crystals

    I. Liquid crystals. On the theory of liquid crystals , author =. Discussions of the Faraday Society , volume =. 1958 , publisher =

    work page 1958

  34. [34]

    Journal of Computational Physics , volume =

    Energy-decreasing exponential time differencing Runge--Kutta methods for phase-field models , author =. Journal of Computational Physics , volume =. 2022 , publisher =

    work page 2022

  35. [35]

    Mathematics of Computation , volume =

    Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows , author =. Mathematics of Computation , volume =

    work page

  36. [36]

    and Stark, H

    Fukuda, J. and Stark, H. and Yoneya, M. and Yokoyama, H. , title =. Phys. Rev. E , volume =

    work page

  37. [37]

    Advanced Materials , volume =

    Unraveling the mystery of the blue fog: structure, properties, and applications of amorphous blue phase III , author =. Advanced Materials , volume =. 2017 , publisher =

    work page 2017

  38. [38]

    Nature Materials , volume =

    Robust cholesteric liquid crystal elastomer fibres for mechanochromic textiles , author =. Nature Materials , volume =. 2022 , publisher =

    work page 2022

  39. [39]

    and Montero, J

    Golovaty, D. and Montero, J. , title =. Arch. Ration. Mech. Anal. , volume =

    work page

  40. [40]

    Journal of Materials Chemistry , volume =

    Chirality in liquid crystals , author =. Journal of Materials Chemistry , volume =. 1991 , publisher =

    work page 1991

  41. [41]

    Journal of Mathematical Physics , year =

    Ambrosio, Luigi and Gigli, Nicola and Savar\'e, Giuseppe and Smith, John , title =. Journal of Mathematical Physics , year =

    work page

  42. [42]

    Physical Review A , volume =

    Landau theory of cholesteric blue phases , author =. Physical Review A , volume =. 1983 , publisher =

    work page 1983

  43. [43]

    Physical Review A , volume =

    Landau theory of cholesteric blue phases: The role of higher harmonics , author =. Physical Review A , volume =. 1984 , publisher =

    work page 1984

  44. [44]

    Numerische Mathematik , volume =

    A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation , author =. Numerische Mathematik , volume =. 2014 , publisher =

    work page 2014

  45. [45]

    Journal of Computational Physics , volume =

    Second order convex splitting schemes for periodic nonlocal Cahn--Hilliard and Allen--Cahn equations , author =. Journal of Computational Physics , volume =. 2014 , publisher =

    work page 2014

  46. [46]

    Macromolecular Chemistry and Physics , volume =

    Intelligent Optical Materials Based on Polymer Dispersed Cholesteric Liquid Crystals , author =. Macromolecular Chemistry and Physics , volume =. 2025 , publisher =

    work page 2025

  47. [47]

    Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions , author =. Adv. Differential Equations , volume =

    work page

  48. [48]

    Physical review letters , volume =

    Structure of blue phase III of cholesteric liquid crystals , author =. Physical review letters , volume =. 2011 , publisher =

    work page 2011

  49. [49]

    Proceedings of the Royal Society A , volume =

    Modelling and simulation of the cholesteric Landau-de Gennes model , author =. Proceedings of the Royal Society A , volume =. 2024 , publisher =

    work page 2024

  50. [50]

    Mathematics of Computation , volume =

    A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility , author =. Mathematics of Computation , volume =

    work page

  51. [51]

    Journal of Display Technology , volume =

    Dual-view blue phase liquid crystal display , author =. Journal of Display Technology , volume =. 2015 , publisher =

    work page 2015

  52. [52]

    and Wise, S

    Hu, Z. and Wise, S. M. and Wang, C. and Lowengrub, J. S. , title =. Journal of Computational Physics , year =

    work page

  53. [53]

    Communications in Computational Physics , volume =

    On the disclination lines of nematic liquid crystals , author =. Communications in Computational Physics , volume =. 2016 , publisher =

    work page 2016

  54. [54]

    2025 , eprint =

    Maximum bound principle for Q-tensor gradient flow with low regularity integrators , author =. 2025 , eprint =

    work page 2025

  55. [55]

    2024 , eprint =

    Error estimate for the first order energy stable scheme of Q-tensor nematic model , author =. 2024 , eprint =

    work page 2024

  56. [56]

    Global Well-Posedness for the Dynamical

    Huang, JinRui and Ding, ShiJin , year =. Global Well-Posedness for the Dynamical. Science China Mathematics , volume =. doi:10.1007/s11425-015-4990-8 , urldate =

    work page doi:10.1007/s11425-015-4990-8

  57. [57]

    and Xu, X

    Iyer, G. and Xu, X. and Zarnescu, A. , title =. Math. Models and Methods Appl. Sci. , volume =

    work page

  58. [58]

    Numerical Methods for Partial Differential Equations , volume =

    Unconditionally stable exponential time differencing schemes for the mass-conserving Allen--Cahn equation with nonlocal and local effects , author =. Numerical Methods for Partial Differential Equations , volume =. 2022 , publisher =

    work page 2022

  59. [59]

    Journal of Scientific Computing , volume =

    Fast explicit integration factor methods for semilinear parabolic equations , author =. Journal of Scientific Computing , volume =. 2015 , publisher =

    work page 2015

  60. [60]

    and Li, X

    Ju, L. and Li, X. and Qiao, Z. and Zhang, H. , title =. Mathematics of Computation , year =

    work page

  61. [61]

    Mathematics of Computation , volume =

    Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection , author =. Mathematics of Computation , volume =

    work page

  62. [62]

    and Bahr, C

    Kitzerow, H.-S. and Bahr, C. , title =

    work page

  63. [63]

    Molecular crystals and liquid crystals , volume =

    The effect of electric fields on blue phases , author =. Molecular crystals and liquid crystals , volume =. 1991 , publisher =

    work page 1991

  64. [64]

    Advanced Materials , volume =

    Chirality of Liquid Crystals Formed from Achiral Molecules Revealed by Resonant X-Ray Scattering , author =. Advanced Materials , volume =. 2020 , publisher =

    work page 2020

  65. [65]

    Communications in Computational Physics , volume =

    Convergence analysis of exponential time differencing schemes for the Cahn-Hilliard equation , author =. Communications in Computational Physics , volume =

    work page

  66. [66]

    and Ju, L

    Li, X. and Ju, L. and Hoang, T. T. P. , title =. BIT Numerical Mathematics , year =

    work page

  67. [67]

    Advances in Computational Mathematics , volume =

    Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation , author =. Advances in Computational Mathematics , volume =. 2020 , publisher =

    work page 2020

  68. [68]

    and Li, X

    Li, J. and Li, X. and Ju, L. and Feng, X. , title =. SIAM Journal on Scientific Computing , year =

    work page

  69. [69]

    Mathematics of computation , volume =

    Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn--Hilliard equation , author =. Mathematics of computation , volume =

    work page

  70. [70]

    and Li, X

    Li, J. and Li, X. and Ju, L. , title =. Numerical Algorithms , year =

    work page

  71. [71]

    IMA journal of numerical analysis , volume =

    Stabilization parameter analysis of a second-order linear numerical scheme for the nonlocal Cahn--Hilliard equation , author =. IMA journal of numerical analysis , volume =. 2023 , publisher =

    work page 2023

  72. [72]

    A Fully Discrete Low-Regularity Integrator for the

    Li, Buyang and Wu, Yifei , year =. A Fully Discrete Low-Regularity Integrator for the. Numerische Mathematik , volume =. doi:10.1007/s00211-021-01226-3 , urldate =

    work page doi:10.1007/s00211-021-01226-3

  73. [73]

    Li, Buyang and Ma, Shu and Schratz, Katharina , year =. A. SIAM Journal on Numerical Analysis , volume =. doi:10.1137/21M1437007 , urldate =

    work page doi:10.1137/21m1437007

  74. [74]

    Stability and Convergence of

    Li, Dong and Quan, Chaoyu and Xu, Jiao , year =. Stability and Convergence of. arXiv , langid =:2108.11254 , primaryclass =

    work page arXiv

  75. [75]

    Archive for Rational Mechanics and Analysis , volume =

    The Oseen--Frank limit of Onsager’s molecular theory for liquid crystals , author =. Archive for Rational Mechanics and Analysis , volume =. 2018 , publisher =

    work page 2018

  76. [76]

    IMA Journal of Numerical Analysis , pages =

    On the maximum bound principle and energy dissipation of exponential time differencing methods for the matrix-valued Allen--Cahn equation , author =. IMA Journal of Numerical Analysis , pages =. 2024 , publisher =

    work page 2024

  77. [77]

    2024 , journal =

    Luo, Daoxing and Wu, Jinbing and Guo, Zhenghao and Hu, Wei and. 2024 , journal =. doi:10.37188/CJLCD.2024-0043 , urldate =

    work page doi:10.37188/cjlcd.2024-0043 2024

  78. [78]

    Quarterly of Applied Mathematics , year =

    Majumdar, Apala and Zarnescu, Arghir , title =. Quarterly of Applied Mathematics , year =

    work page

  79. [79]

    European Journal of Applied Mathematics , volume =

    Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory , author =. European Journal of Applied Mathematics , volume =. 2010 , publisher =

    work page 2010

  80. [80]

    Archive for rational mechanics and analysis , volume =

    Landau--de Gennes theory of nematic liquid crystals: the Oseen--Frank limit and beyond , author =. Archive for rational mechanics and analysis , volume =. 2010 , publisher =

    work page 2010

Showing first 80 references.