On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation

• Published in: Advances in computational mathematics Vol. 50; no. 3
• Main Authors: Zhang, Hong, Zhang, Gengen, Liu, Ziyuan, Qian, Xu, Song, Songhe
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V
• Subjects: Accuracy; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Convergence; Integrators; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Maximum principle; Rescaling; Runge-Kutta method; Time dependence; Time lag; Visualization; 65L06; Parametric Runge–Kutta method; Forward Euler condition; Maximum principle; 35B50; Time-step relaxation; 65M12; Viscous Cahn–Hilliard equation; 35K55
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10143-6

The stabilization approach has been known to permit large time-step sizes while maintaining stability. However, it may “slow down the convergence rate” or cause “delayed convergence” if the time-step rescaling is not well resolved. By considering a fourth-order-in-space viscous Cahn–Hilliard (VCH) equation, we propose a class of up to the fourth-order single-step methods that are able to capture the correct physical behaviors with high-order accuracy and without time delay. By reformulating the VCH as a system consisting of a second-order diffusion term and a nonlinear term involving the operator ( I - ν Δ ) - 1 , we first develop a general approach to estimate the maximum bound for the VCH equation equipped with either the Ginzburg–Landau or Flory–Huggins potential. Then, by taking advantage of new recursive approximations and adopting a time-step-dependent stabilization, we propose a class of stabilization Runge–Kutta methods that preserve the maximum principle for any time-step size without harming the convergence. Finally, we transform the stabilization method into a parametric Runge–Kutta formulation, estimate the rescaled time-step, and remove the time delay by means of a relaxation technique. When the stabilization parameter is chosen suitably, the proposed parametric relaxation integrators are rigorously proven to be mass-conserving, maximum-principle-preserving, and the convergence in the l ∞ -norm is estimated with p th-order accuracy under mild regularity assumption. Numerical experiments on multi-dimensional benchmark problems are carried out to demonstrate the stability, accuracy, and structure-preserving properties of the proposed schemes.