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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (1): 120-134.doi: 10.3969/j.issn.1005-3085.2022.01.009

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求解带有不连续波数的二维变系数 Helmholtz 方程的一种高精度紧致差分方法

王  芳,   冯秀芳   

  1. 宁夏大学数学统计学院，银川 750021
  • 出版日期:2022-02-15 发布日期:2022-04-15
  • 基金资助:
    国家自然科学基金 (11961054)；宁夏自然科学基金 (2020AAC03069).

A High-order Compact Difference Method for Solving Two-dimensional Variable Coefficients Helmholtz Equation with Discontinuous Wave Number

WANG Fang,  FENG Xiufang   

  1. School of Mathematical Statistics, Ningxia University, Yinchuan 750021
  • Online:2022-02-15 Published:2022-04-15
  • Supported by:
    The National Natural Science Foundation of China (11961054); the Natural Science Foundation of Ningxia (2020AAC03069).

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摘要：

很多实际物理问题都可以由带有不连续波数的变系数 Helmholtz 方程进行数值模拟。Helmholtz 方程的数值方法研究是热点问题之一，具有重要的理论和实际意义。由于波数的不连续性，使用传统的有限差分方法求解带有不连续波数的 Helmholtz 方程时通常无法达到原有差分格式的精度。结合浸入界面方法的思想，对带有不连续波数的二维变系数 Helmholtz 方程构造了一类新的四阶紧致有限差分格式，数值实验验证了新方法的可靠性和有效性。

关键词: 变系数 Helmhlotz 方程, 浸入界面方法, 紧致格式, 有限差分方法

Abstract:

Many practical physical problems can be numerically simulated by the variable coefficients Helmholtz equation with discontinuous wave number. The numerical method of the Helmholtz equation, with important theoretical and practical significance, is one of the hot research topics. Due to the discontinuity of wave number, the traditional finite difference method is usually unable to achieve the accuracy of the original difference scheme when solving Helmholtz equation with discontinuous wave number. Based on the idea of immersed interface method, a new fourth-order compact finite difference scheme is constructed  for two-dimensional coefficient Helmholtz equation with discontinuous wave number. The reliability and effectiveness of the new method are verified by numerical experiments.

Key words: variable coefficients Helmhlotz equation, immersed interface method, compact scheme, finite difference method

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