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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]
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Database: Engineering Source
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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]
ISSN:09615539