变密度Ericksen-Leslie方程的高效数值算法及误差分析  

Efficient Numerical Algorithm and Error Analysis for the Ericksen-Leslie Equations with Variable Density

作　　者：张鑫 王旦霞 张建文 贾宏恩 ZHANG Xin;WANG Danxia;ZHANG Jianwen;JIA Hongen(School of Mathematics,Taiyuan University of Technology,Jinzhong Shanxi 030600;Shanxi Key Laboratory for Intelligent Optimization Computing and Blockchain Technology,Taiyuan Shanxi 030000,China)

机构地区：[1]太原理工大学数学学院,山西晋中030600 [2]智能优化计算与区块链技术山西省重点实验室,太原030000

出　　处：《重庆师范大学学报（自然科学版）》2024年第3期79-88,共10页Journal of Chongqing Normal University:Natural Science

基　　金：山西省科技合作交流专项项目(No.202104041101019);山西省回国留学人员科研资助项目(No.2021-029);山西省基础研究计划(No.202203021211129);国家自然科学基金面上项目(No.11872264)。

摘　　要：针对变密度Ericksen-Leslie方程提出了一种高效的数值算法。首先,通过自由能定义一个标量辅助变量(scalar auxiliary variable, SAV)并由此得到一个等价的新系统。其次,对新系统建立一个数值格式,其中Ginzburg-Landau惩罚函数通过SAV被显式处理从而将非线性项线性化。理论分析证明了格式的唯一可解性和无条件能量稳定性,并且通过严格的误差分析证明了格式的一阶收敛率。通过数值模拟验证了理论推导结果,并给出了奇点湮灭过程。所构造的格式在理论上和数值计算中都保持了预期的精度,并且在演化模拟中表现出良好的性能。An efficient numerical algorithm is proposed for the Ericksen-Leslie equations with variable density.Firstly,by introducing a SAV(scalar auxiliary variable)with the free energy,an equivalent new system is obtained.Secondly,a numerical scheme of the new system is established where the Ginzburg-Landau penalty function is handled explicitly such that the nonlinear terms are linearized.Theoretical analysis has demonstrated the unique solvability and unconditional energy stability of the scheme,and the first-order convergence rate of the scheme has been proved by rigorous error analysis.The theoretical derivation results were verified by several numerical simulations,and the singularity annihilation process was presented.The constructed scheme maintains the expected accuracy in both theoretical and numerical calculations,and exhibits good performance in evolutionary simulations.

关 键 词：Ericksen-Leslie方程 变密度 SAV 无条件能量稳定 误差分析 

分 类 号：O241[理学—计算数学]