Error estimates of linear decoupled structure-preserving incremental viscosity splitting methods for the Cahn--Hilliard--Navier--Stokes system
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arXiv
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| Format: | Preprint |
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2025
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| author | Kuang, Baolin Fu, Hongfei Li, Xiaoli |
| author_facet | Kuang, Baolin Fu, Hongfei Li, Xiaoli |
| contents | We propose first- and second-order time discretization schemes for the coupled Cahn--Hilliard--Navier--Stokes model, leveraging the incremental viscosity splitting (IVS) method. The schemes combine the scalar auxiliary variable method and the zero-energy-contribution approach, resulting in a linear, decoupled numerical framework. At each time step, they only require to solve a sequence of constant-coefficient equations, along with a linear equation with one unknown, making the algorithms computationally efficient and easy to implement. In addition, the proposed schemes are proven to be uniquely solvable, mass-conserving, and unconditional energy dissipation. Most importantly, leveraging the mathematical induction method and the regularity properties of the Stokes equation, we perform a rigorous error analysis for the first-order scheme in multiple space dimensions, establishing an unconditional and optimal convergence rate for all relevant variables under different norms. A user-defined, time-dependent parameter plays an important role in the error analysis of the proposed structure-preserving IVS methods. Ample numerical examples are carried out to verify the theoretical findings and to demonstrate the accuracy, effectiveness and efficiency of the proposed schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error estimates of linear decoupled structure-preserving incremental viscosity splitting methods for the Cahn--Hilliard--Navier--Stokes system Kuang, Baolin Fu, Hongfei Li, Xiaoli Numerical Analysis 35Q35, 65M12, 65M15, 76D05 We propose first- and second-order time discretization schemes for the coupled Cahn--Hilliard--Navier--Stokes model, leveraging the incremental viscosity splitting (IVS) method. The schemes combine the scalar auxiliary variable method and the zero-energy-contribution approach, resulting in a linear, decoupled numerical framework. At each time step, they only require to solve a sequence of constant-coefficient equations, along with a linear equation with one unknown, making the algorithms computationally efficient and easy to implement. In addition, the proposed schemes are proven to be uniquely solvable, mass-conserving, and unconditional energy dissipation. Most importantly, leveraging the mathematical induction method and the regularity properties of the Stokes equation, we perform a rigorous error analysis for the first-order scheme in multiple space dimensions, establishing an unconditional and optimal convergence rate for all relevant variables under different norms. A user-defined, time-dependent parameter plays an important role in the error analysis of the proposed structure-preserving IVS methods. Ample numerical examples are carried out to verify the theoretical findings and to demonstrate the accuracy, effectiveness and efficiency of the proposed schemes. |
| title | Error estimates of linear decoupled structure-preserving incremental viscosity splitting methods for the Cahn--Hilliard--Navier--Stokes system |
| topic | Numerical Analysis 35Q35, 65M12, 65M15, 76D05 |
| url | https://arxiv.org/abs/2508.01141 |