arXiv subjects

Haijun Gao

Publications and source records attributed to Haijun Gao.

5 recordsLinked to original sources

arXiv2025

Second-order unconditionally stable time-filtered scheme for Cahn-Hilliard-Navier-Stokes system

In this work, we introduce the time filtering technique to develop several innovative semi-discrete schemes in time for the Cahn-Hilliard-Navier-Stokes (CHNS) system. These schemes achieve second-order temporal accuracy while maintaining unconditional energy stability. Our approach begins with the discretization of the CHNS system using the first-order semi-implicit method. Subsequently, by applying time filtering techniques, we improve the temporal accuracy from first-order to second-order. This improvement requires only minor modifications to the original first-order semi-implicit scheme, thereby enabling higher accuracy to be achieved at minimal cost. Moreover, we rigorously establish the unconditional energy stability of the proposed schemes through theoretical analysis. Additionally, we extend our work to develop semi-discrete schemes that incorporate variable and adaptive time-stepping strategies, enhancing the flexibility and efficiency of simulations. Numerical examples are presented to validate the theoretical results and demonstrate the effectiveness of the proposed methods.

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arXiv2025

Optimal convergence analysis of fully discrete SAVs-FEM for the Cahn-Hilliard-Navier-Stokes equations

We construct a fully discrete numerical scheme that is linear, decoupled, and unconditionally energy stable, and analyze its optimal error estimates for the Cahn-Hilliard-Navier-Stokes equations. For time discretization, we employ the two scalar auxiliary variables (SAVs) and the pressure-correction projection method. For spatial discretization, we choose the $P_r \times P_r \times \mathbf{P}_{r+1} \times P_r$ finite element spaces, where $r$ is the degree of the local polynomials, and derive the optimal $L^2$ error estimates for the phase-field variable, chemical potential, and pressure in the case of $r \geq 1$, and for the velocity when $r \geq 2$, without relying on the quasi-projection operator technique proposed in \textit{[Cai et al. SIAM J Numer Anal, 2023]}. Numerical experiments validate the theoretical results, confirming the unconditional energy stability and optimal convergence rates of the proposed scheme. Additionally, we numerically demonstrate the optimal $L^2$ convergence rate for the velocity when $r=1$.

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arXiv2025

Convergence analysis of decoupled mixed FEM for the Cahn-Hilliard-Navier-Stokes equations

We develop a decoupled, first-order, fully discrete, energy-stable scheme for the Cahn-Hilliard-Navier-Stokes equations. This scheme calculates the Cahn-Hilliard and Navier-Stokes equations separately, thus effectively decoupling the entire system. To further separate the velocity and pressure components in the Navier-Stokes equations, we use the pressure-correction projection method. We demonstrate that the scheme is primitively energy stable and prove the optimal $L^2$ error estimate of the fully discrete scheme in the $P_r\times P_r\times P_r\times P_{r-1}$ finite element spaces, where the phase field, chemical potential, velocity and pressure satisfy the first-order accuracy in time and the $\left(r+1,r+1,r+1,r\right)th$-order accuracy in space, respectively. Furthermore, numerical experiments are conducted to support these theoretical findings. Notably, compared to other numerical schemes, our algorithm is more time-efficient and numerically shown to be unconditionally stable.

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arXiv2024

A new decoupled unconditionally stable scheme and its optimal error analysis for the Cahn-Hilliard-Navier-Stokes equations

We construct a decoupled, first-order, fully discrete, and unconditionally energy stable scheme for the Cahn-Hilliard-Navier-Stokes equations. The scheme is divided into two main parts. The first part involves the calculation of the Cahn-Hilliard equations, and the other part is calculating the Navier-Stokes equations subsequently by utilizing the phase field and chemical potential values obtained from the above step. Specifically, the velocity in the Cahn-Hilliard equation is discretized explicitly at the discrete time level, which enables the computation of the Cahn-Hilliard equations is fully decoupled from that of Navier-Stokes equations. Furthermore, the pressure-correction projection method, in conjunction with the scalar auxiliary variable approach not only enables the discrete scheme to satisfy unconditional energy stability, but also allows the convective term in the Navier-Stokes equations to be treated explicitly. We subsequently prove that the time semi-discrete scheme is unconditionally stable and analyze the optimal error estimates for the fully discrete scheme. Finally, several numerical experiments validate the theoretical results.

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arXiv2021

Covariance Self-Attention Dual Path UNet for Rectal Tumor Segmentation

Deep learning algorithms are preferable for rectal tumor segmentation. However, it is still a challenge task to accurately segment and identify the locations and sizes of rectal tumors by using deep learning methods. To increase the capability of extracting enough feature information for rectal tumor segmentation, we propose a Covariance Self-Attention Dual Path UNet (CSA-DPUNet). The proposed network mainly includes two improvements on UNet: 1) modify UNet that has only one path structure to consist of two contracting path and two expansive paths (nam new network as DPUNet), which can help extract more feature information from CT images; 2) employ the criss-cross self-attention module into DPUNet, meanwhile, replace the original calculation method of correlation operation with covariance operation, which can further enhances the characterization ability of DPUNet and improves the segmentation accuracy of rectal tumors. Experiments illustrate that compared with the current state-of-the-art results, CSA-DPUNet brings 15.31%, 7.2%, 11.8%, and 9.5% improvement in Dice coefficient, P, R, F1, respectively, which demonstrates that our proposed CSA-DPUNet is effective for rectal tumor segmentation.

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