海面下方多障碍体散射问题数值研究  

NUMERICAL STUDY ON A SCATTERING of MULTIPLE OBSTACLES BELOW SEA SURFACE

作　　者：王珏[1] 亓艳 Wang Jue;Qi Yan(School of Mathematics,Hangzhou Normal University,Hangzhou 311121,China;School of Mathematical Sciences,Harbin Engineering University,Harbin 150001,China)

机构地区：[1]杭州师范大学数学学院,杭州311121 [2]哈尔滨工程大学数学科学学院,哈尔滨150001

出　　处：《计算数学》2024年第1期47-78,共32页Mathematica Numerica Sinica

基　　金：国家自然科学基金项目(12371420);浙江省自然科学基金项目(LY23A010004);科研启动基金项目(2022QDL017)资助。

摘　　要：本文针对二维空间中海面下方多障碍体散射问题,分别从理论分析和数值计算两方面进行研究.通过分析散射问题的特性,利用Helmholtz方程,结合不同边界条件以及无穷远处辐射条件,建立了海面下方多障碍体散射问题的数学模型,并证明了散射问题解的唯一性.基于位势理论,利用间接积分方程方法,得到了不同区域的场所满足的积分表示,以及边界上密度函数所满足的边界积分方程.通过引入位势算子,将积分区域进行截断,得到有界域上的算子方程.针对所建立的边界积分方程系统,利用Nystr?m方法构造数值格式,并证明了数值解的收敛性.最后,利用数值实验验证理论的正确性和有效性.进一步,通过设计数值实验分析不同参数对散射问题的影响.In this paper,the scattering problem of multiple obstacles under the sea surface in twodimensional space is studied theoretically and numerically.By analyzing the characteristics of the scattering problem,using the Helmholtz equation,and combining different boundary conditions and radiation conditions,the mathematical model is established,and the uniqueness of the scattering problem is proved.Based on the potential theory and the indirect integral equation method,the integral representation of the fields in different regions and the integral boundary equation of the density function on the boundary is obtained.By introducing potential operator,the integral domain is truncated,and the operator equation on the bounded domain is obtained.For the established boundary integral equation system,the numerical scheme is constructed using the Nystrom method,and the convergence of the numerical solution is proved.Finally,numerical experiments are used to verify the correctness and effectiveness of the theory.Furthermore,numerical experiments are designed to analyze the effects of different parameters on the scattering problem.

关 键 词：积分方程方法 散射问题 唯一性 位势理论 数值分析 

分 类 号：O241.83[理学—计算数学]