View in EDS

A multigrid solver for the Allen–Cahn equation on a virtual cubic surface.

Saved in:
Bibliographic Details
Title: A multigrid solver for the Allen–Cahn equation on a virtual cubic surface.
Authors: Yang, Junxiang1 (AUTHOR), Kwak, Soobin2 (AUTHOR), Ham, Seokjun2 (AUTHOR), Hwang, Youngjin2 (AUTHOR), Kim, Hyundong3,4 (AUTHOR), Ma, Juho2 (AUTHOR), Kim, Junseok1,2 (AUTHOR) cfdkim@korea.ac.kr
Source: Applied Numerical Mathematics. Oct2026, Vol. 228, p71-96. 26p.
Subjects: Multigrid methods (Numerical analysis), Geometric surfaces, Phase transitions, Reaction-diffusion equations, Stability (Mechanics), Finite difference method, Heat equation, Interface dynamics
Abstract: We present an unconditionally stable hybrid finite difference scheme with a multigrid solver for the nonlinear Allen–Cahn (AC) model on a virtual cubic surface. Previous fully explicit operator splitting methods suffer from strict time step restrictions that limit computational efficiency. To overcome this, the diffusion term is discretized implicitly and solved using a V-cycle multigrid algorithm, while the nonlinear term is updated analytically in closed form. The proposed scheme incorporates boundary conditions reflecting the virtual cubic surface geometry and achieves second-order spatial accuracy and first-order temporal accuracy. Numerical tests confirm that the proposed algorithm preserves the discrete maximum principle and satisfies the energy dissipation property within a multigrid tolerance, even for large time steps beyond the explicit stability limit. Computational experiments include convergence analysis, validation of energy and maximum principle preservation, and simulations of motion by mean curvature and traveling wave propagation on the cubic surface. Computational results demonstrate that the scheme accurately captures interface dynamics, including curvature-driven motion and complex shape evolution, and maintains both stability and efficiency. The use of the multigrid method ensures fast convergence for the diffusion step, and the approach is well suited for large-scale simulations of phase-field models on curved or piecewise-planar surfaces. This work provides a practical and robust computational methodology for solving the nonlinear AC equation on the virtual surfaces of a rectangular cuboid, and it enables efficient simulation of interfacial phenomena with reduced computational cost and without restrictive time step constraints. [ABSTRACT FROM AUTHOR]
Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
FullText Text:
  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 194254089
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: A multigrid solver for the Allen–Cahn equation on a virtual cubic surface.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Yang%2C+Junxiang%22">Yang, Junxiang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kwak%2C+Soobin%22">Kwak, Soobin</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Ham%2C+Seokjun%22">Ham, Seokjun</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Hwang%2C+Youngjin%22">Hwang, Youngjin</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kim%2C+Hyundong%22">Kim, Hyundong</searchLink><relatesTo>3,4</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Ma%2C+Juho%22">Ma, Juho</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kim%2C+Junseok%22">Kim, Junseok</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> cfdkim@korea.ac.kr</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22Applied+Numerical+Mathematics%22">Applied Numerical Mathematics</searchLink>. Oct2026, Vol. 228, p71-96. 26p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Multigrid+methods+%28Numerical+analysis%29%22">Multigrid methods (Numerical analysis)</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+surfaces%22">Geometric surfaces</searchLink><br /><searchLink fieldCode="DE" term="%22Phase+transitions%22">Phase transitions</searchLink><br /><searchLink fieldCode="DE" term="%22Reaction-diffusion+equations%22">Reaction-diffusion equations</searchLink><br /><searchLink fieldCode="DE" term="%22Stability+%28Mechanics%29%22">Stability (Mechanics)</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+difference+method%22">Finite difference method</searchLink><br /><searchLink fieldCode="DE" term="%22Heat+equation%22">Heat equation</searchLink><br /><searchLink fieldCode="DE" term="%22Interface+dynamics%22">Interface dynamics</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We present an unconditionally stable hybrid finite difference scheme with a multigrid solver for the nonlinear Allen–Cahn (AC) model on a virtual cubic surface. Previous fully explicit operator splitting methods suffer from strict time step restrictions that limit computational efficiency. To overcome this, the diffusion term is discretized implicitly and solved using a V-cycle multigrid algorithm, while the nonlinear term is updated analytically in closed form. The proposed scheme incorporates boundary conditions reflecting the virtual cubic surface geometry and achieves second-order spatial accuracy and first-order temporal accuracy. Numerical tests confirm that the proposed algorithm preserves the discrete maximum principle and satisfies the energy dissipation property within a multigrid tolerance, even for large time steps beyond the explicit stability limit. Computational experiments include convergence analysis, validation of energy and maximum principle preservation, and simulations of motion by mean curvature and traveling wave propagation on the cubic surface. Computational results demonstrate that the scheme accurately captures interface dynamics, including curvature-driven motion and complex shape evolution, and maintains both stability and efficiency. The use of the multigrid method ensures fast convergence for the diffusion step, and the approach is well suited for large-scale simulations of phase-field models on curved or piecewise-planar surfaces. This work provides a practical and robust computational methodology for solving the nonlinear AC equation on the virtual surfaces of a rectangular cuboid, and it enables efficient simulation of interfacial phenomena with reduced computational cost and without restrictive time step constraints. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=194254089
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1016/j.apnum.2026.05.010
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 26
        StartPage: 71
    Subjects:
      – SubjectFull: Multigrid methods (Numerical analysis)
        Type: general
      – SubjectFull: Geometric surfaces
        Type: general
      – SubjectFull: Phase transitions
        Type: general
      – SubjectFull: Reaction-diffusion equations
        Type: general
      – SubjectFull: Stability (Mechanics)
        Type: general
      – SubjectFull: Finite difference method
        Type: general
      – SubjectFull: Heat equation
        Type: general
      – SubjectFull: Interface dynamics
        Type: general
    Titles:
      – TitleFull: A multigrid solver for the Allen–Cahn equation on a virtual cubic surface.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Yang, Junxiang
      – PersonEntity:
          Name:
            NameFull: Kwak, Soobin
      – PersonEntity:
          Name:
            NameFull: Ham, Seokjun
      – PersonEntity:
          Name:
            NameFull: Hwang, Youngjin
      – PersonEntity:
          Name:
            NameFull: Kim, Hyundong
      – PersonEntity:
          Name:
            NameFull: Ma, Juho
      – PersonEntity:
          Name:
            NameFull: Kim, Junseok
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 10
              Text: Oct2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 01689274
          Numbering:
            – Type: volume
              Value: 228
          Titles:
            – TitleFull: Applied Numerical Mathematics
              Type: main
ResultId 1