Two Second-Order Ecient Numerical Schemes for the Boussinesq Equations  

Boussinesq方程的两种高效二阶数值格式

作　　者：LIU Fang WANG Danxia ZHANG Jianwen 刘芳;王旦霞;张建文(太原理工大学数学学院,山西晋中030600;智能优化计算与区块链技术山西省重点实验室,山西晋中030600)

机构地区：[1]School of Mathematics,Taiyuan University of Technology,Jinzhong 030600,China [2]Shanxi Key Laboratory for Intelligent Optimization Computing and Blockchain Technology,Jinzhong 030600,China

出　　处：《应用数学》2025年第1期114-129,共16页Mathematica Applicata

基　　金：Supported by Research Project Supported by Shanxi Scholarship Council of China(2021-029);International Cooperation Base and Platform Project of Shanxi Province(202104041101019);Basic Research Plan of Shanxi Province(202203021211129);Shanxi Province Natural Science Research(202203021212249);Special/Youth Foundation of Taiyuan University of Technology(2022QN101)。

摘　　要：In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.本文研究Boussinesq方程的高效数值求解问题,首先利用标量辅助变量(SAV)方法,将原Boussinesq方程改写为等价的Boussinesq方程,然后利用二阶后向微分公式(BDF2),Crank-Nicolson(CN)方法和压力校正法对其进行离散,得到两个二阶全解耦的时间离散格式.经过严格的理论分析,得到它们的无条件稳定性,唯一可解性和解耦的详细实现过程.最后,进行了各种二维数值模拟,验证了所提方案的精度和能量稳定性.

关 键 词：Scalar auxiliary variable approach Pressure-correction method Fully decoupled Unconditional stability Boussinesq equations 

分 类 号：O241.82[理学—计算数学]