一维边界阻尼波动方程的三层隐式θ-差分格式及其降维算法  

A Three-Layer Implicitθ-Difference Scheme and Its Dimension Reduction Algorithm for One-Dimensional Boundary Damped Wave Equation

作　　者：马湘荣 李宏 MA Xiangrong;LI Hong(School of Mathematical Sciences,Inner Mongolia University,Hohhot 010020,China)

机构地区：[1]内蒙古大学数学科学学院,内蒙古呼和浩特010020

出　　处：《应用数学》2024年第4期1087-1102,共16页Mathematica Applicata

基　　金：国家自然科学基金(12161063);内蒙古自然科学基金(2021MS01018);内蒙古自治区高等学校创新团队发展计划(NMGIRT2207)。

摘　　要：针对边界阻尼波动方程构造全离散三层隐式加权有限差分θ-格式,证明了当θ<1/2时,具有时间一阶精度和空间二阶精度;当θ=1/2时,关于时间和空间都是二阶精度格式,并证明该格式在θ≤1/2时是无条件稳定的.特别是,利用特征投影分解(POD)法对该格式做降维,建立了θ-格式的POD降维算法,分析了POD降维算法解的稳定性和误差.最后,利用数值算例验证了上述所有理论结果的正确性.In this paper,a fully discrete three-layer implicit weighted finite difference schemeθ-scheme is constructed for the boundary damped wave equation.It is proved that whenθ<12,there is the first-order accuracy in time and the second-order accuracy in space,and whenθ=12,it has the secondorder accuracy with respect to both time and space.It is proved that the scheme is unconditionally stable for■.In particular,the proper orthogonality decomposition(POD)method is used to reduce the dimension of the scheme and the POD-based reduced-dimension algorithm is established.The stability and errors for the POD reduced-dimension algorithm are analyzed.Finally,some numerical example are used to verify the correctness of all the above theoretical results.

关 键 词：边界阻尼波动方程 有限差分格式 收敛性和稳定性 特征投影分解 优化算法 

分 类 号：O241.82[理学—计算数学]