Generalized- <i>α</i> methods for the Navier–Stokes–Cahn–Hilliard equations

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.