摘要
【目的】由于界面问题所导出的偏微分方程的解在通过界面时一般是不连续的,这使得大多数传统数值方法不能很好地适用于求解界面问题,而有限体积方法因保持物理量的局部守恒性,而且计算简单,于是成为解决界面问题的有效方法。因此,研究利用有限体积方法对求解界面问题具有重要意义。【方法】首先基于一种修正的有限体积方法对带有不连续波数和奇异源项的Helmholtz方程进行整体逼近。然后,通量采用泰勒级数展开,积分项利用多项式插值进行逼近,对于界面问题利用跳跃条件将负侧的点转化到正侧,从而构造了连续问题以及界面问题的六阶紧致有限差分格式。【结果】格式在连续波数和界面处都可以达到六阶精度。【结论】数值实验验证了格式的有效性和精确性。
[Purposes]Since the solutions of partial differential equations derived from interface problems are generally discontinuous when pass through the interface,most of the traditional numerical methods are not suitable for solving interface problems,while the finite volume method is effective for solving interface problems because of its local conservation of physical quantities and simple calculation.Therefore,the finite volume method is of great significance to solve the interface problem.[Methods]Firstly,a modified finite volume method is used to approximate the Helmholtz equation with discontinuous wave numbers and singular source terms.Then,the flux is expanded by Taylor series,and the integral term is approximated by polynomial interpolation.For the interface problem,the points on the negative side are transformed to the positive side by using the jump condition.[Findings]The scheme can achieve sixth order accuracy at both continuous wave number and interface.[Conclusions]Numerical experiments verify the effectiveness and accuracy of the scheme.
作者
张娟
冯秀芳
ZHANG Juan;FENG Xiufang(School of Mathematics and Statistics,Ningxia University,Yinchuan 750021,China)
出处
《重庆师范大学学报（自然科学版）》
CAS
北大核心
2021年第4期69-79,共11页
Journal of Chongqing Normal University:Natural Science
基金
国家自然科学基金(No.11961054)
宁夏自然科学基金(No.NZ16011)。
