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Generalized-α methods for the Navier–Stokes–Cahn–Hilliard equations.

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Title: Generalized-α methods for the Navier–Stokes–Cahn–Hilliard equations.
Authors: van Sluijs, Tom1 (AUTHOR) t.b.v.sluijs@tue.nl, Stoter, Stein1 (AUTHOR) k.f.s.stoter@tue.nl, Behnoudfar, Pouria2 (AUTHOR) behnoudfar@wisc.edu, van Brummelen, Harald1 (AUTHOR) e.h.v.brummelen@tue.nl
Source: International Journal of Numerical Methods for Heat & Fluid Flow. 2026, Vol. 36 Issue 7, p2460-2482. 23p.
Subjects: Time integration scheme, Multiphase flow, Partial differential equations, Computer simulation
Abstract: Purpose: Diffuse-interface models provide a versatile framework for simulating binary-fluid flows with complex interfacial dynamics, including topological changes and dynamic wetting. Numerical approximation of the underlying Navier–Stokes–Cahn–Hilliard (NSCH) equations is challenging due to spatiotemporal multiscale behavior, ε -conditional stability and ill-conditioning. The purpose of this work is to evaluate higher-order generalized-α time-integration methods, focusing on the third-order scheme, for approximating the time-evolution of the NSCH equations. Design/methodology/approach: The authors regard the application of higher-order generalized-α methods to the NSCH system. Their two-step single-stage form facilitates temporally varying spatial adaptivity, providing a framework for effectively resolving the spatiotemporal multiscale behavior of the NSCH equations. In addition, generalized-α schemes offer tunable numerical dissipation and built-in error estimates for adaptive time-stepping strategies. A one-dimensional numerical experiment is presented to elucidate the properties of the generalized-α scheme for the NSCH equations and to compare its performance to classical θ-methods. Findings: The generalized-α scheme attains its theoretical asymptotic convergence rate and high accuracy at small time-step sizes, outperforming classical θ-methods. However, at larger time steps, the accuracy of the method deteriorates and nonlinear instabilities arise. Moreover, higher-order generalized-α schemes entail substantial algorithmic complexity for NSCH systems, particularly with non-matching densities and viscosities, due to the systems' many complex nonlinearities. Originality/value: Most investigations of time integrators for NSCH systems are limited to first- or second-order schemes. This work is the first to examine higher-order generalized-α methods for the NSCH system exhibiting both their advantages and limitations and providing valuable insights into trade-offs between accuracy, stability, efficiency, versatility and algorithmic complexity of the generalized-α scheme relative to classical time-integration methods. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Numerical Methods for Heat & Fluid Flow is the property of Emerald Publishing Limited 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
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DbLabel: Engineering Source
An: 194733976
AccessLevel: 6
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  Data: Generalized-α methods for the Navier–Stokes–Cahn–Hilliard equations.
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  Data: <searchLink fieldCode="AR" term="%22van+Sluijs%2C+Tom%22">van Sluijs, Tom</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> t.b.v.sluijs@tue.nl</i><br /><searchLink fieldCode="AR" term="%22Stoter%2C+Stein%22">Stoter, Stein</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> k.f.s.stoter@tue.nl</i><br /><searchLink fieldCode="AR" term="%22Behnoudfar%2C+Pouria%22">Behnoudfar, Pouria</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> behnoudfar@wisc.edu</i><br /><searchLink fieldCode="AR" term="%22van+Brummelen%2C+Harald%22">van Brummelen, Harald</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> e.h.v.brummelen@tue.nl</i>
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Numerical+Methods+for+Heat+%26+Fluid+Flow%22">International Journal of Numerical Methods for Heat & Fluid Flow</searchLink>. 2026, Vol. 36 Issue 7, p2460-2482. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Time+integration+scheme%22">Time integration scheme</searchLink><br /><searchLink fieldCode="DE" term="%22Multiphase+flow%22">Multiphase flow</searchLink><br /><searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Computer+simulation%22">Computer simulation</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Purpose: Diffuse-interface models provide a versatile framework for simulating binary-fluid flows with complex interfacial dynamics, including topological changes and dynamic wetting. Numerical approximation of the underlying Navier–Stokes–Cahn–Hilliard (NSCH) equations is challenging due to spatiotemporal multiscale behavior, ε -conditional stability and ill-conditioning. The purpose of this work is to evaluate higher-order generalized-α time-integration methods, focusing on the third-order scheme, for approximating the time-evolution of the NSCH equations. Design/methodology/approach: The authors regard the application of higher-order generalized-α methods to the NSCH system. Their two-step single-stage form facilitates temporally varying spatial adaptivity, providing a framework for effectively resolving the spatiotemporal multiscale behavior of the NSCH equations. In addition, generalized-α schemes offer tunable numerical dissipation and built-in error estimates for adaptive time-stepping strategies. A one-dimensional numerical experiment is presented to elucidate the properties of the generalized-α scheme for the NSCH equations and to compare its performance to classical θ-methods. Findings: The generalized-α scheme attains its theoretical asymptotic convergence rate and high accuracy at small time-step sizes, outperforming classical θ-methods. However, at larger time steps, the accuracy of the method deteriorates and nonlinear instabilities arise. Moreover, higher-order generalized-α schemes entail substantial algorithmic complexity for NSCH systems, particularly with non-matching densities and viscosities, due to the systems' many complex nonlinearities. Originality/value: Most investigations of time integrators for NSCH systems are limited to first- or second-order schemes. This work is the first to examine higher-order generalized-α methods for the NSCH system exhibiting both their advantages and limitations and providing valuable insights into trade-offs between accuracy, stability, efficiency, versatility and algorithmic complexity of the generalized-α scheme relative to classical time-integration methods. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of International Journal of Numerical Methods for Heat & Fluid Flow is the property of Emerald Publishing Limited 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.)
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RecordInfo BibRecord:
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    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 23
        StartPage: 2460
    Subjects:
      – SubjectFull: Time integration scheme
        Type: general
      – SubjectFull: Multiphase flow
        Type: general
      – SubjectFull: Partial differential equations
        Type: general
      – SubjectFull: Computer simulation
        Type: general
    Titles:
      – TitleFull: Generalized-α methods for the Navier–Stokes–Cahn–Hilliard equations.
        Type: main
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          Name:
            NameFull: van Sluijs, Tom
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            NameFull: Stoter, Stein
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            NameFull: Behnoudfar, Pouria
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            NameFull: van Brummelen, Harald
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          Dates:
            – D: 01
              M: 07
              Text: 2026
              Type: published
              Y: 2026
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              Value: 36
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              Value: 7
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            – TitleFull: International Journal of Numerical Methods for Heat & Fluid Flow
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