ORIGINAL PAPER
Error estimates of VBDF2 numerical scheme for Cahn–Hilliard–Navier–Stokes model based on IEQ method
Wenzhuo Chen,
Danxia Wang,
Hongen Jia,
Jun Zhang,
Wenzhuo Chen
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, China
Search for more papers by this authorCorresponding Author
Danxia Wang
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, China
Correspondence
Danxia Wang, School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China.
Email: [email protected]
Search for more papers by this authorHongen Jia
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, China
Search for more papers by this authorJun Zhang
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou, China
Search for more papers by this authorWenzhuo Chen,
Danxia Wang,
Hongen Jia,
Jun Zhang,
Wenzhuo Chen
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, China
Search for more papers by this authorCorresponding Author
Danxia Wang
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, China
Correspondence
Danxia Wang, School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China.
Email: [email protected]
Search for more papers by this authorHongen Jia
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, China
Search for more papers by this authorJun Zhang
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou, China
Search for more papers by this authorFirst published: 15 July 2025
Abstract
In this paper, we successfully propose a highly efficient linear, decoupled, fully discrete, variable time step second-order backward difference formula (VBDF2) finite element numerical scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) model. First, by utilizing the invariant energy quadratization (IEQ) method, we derive the equivalent CHNS model, which is crucial for handling the nonlinear terms. Second, by employing the finite element method for spatial discretization and the VBDF2 approach for time discretization, we obtain the fully discrete numerical scheme. Then, we also demonstrate the unique solvability, mass conservation property, and stability of the IEQ-VBDF2 scheme. Subsequently, we present the error estimates accompanied by rigorous theoretical derivations. Finally, we creatively design an adaptive variable-time-step strategy and conduct several numerical simulations to verify the theoretical results. This strategy boasts excellent performance. It can maintain a high level of computational accuracy while significantly enhancing computational efficiency, offering a more efficient and reliable solution for the relevant numerical computing field.
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