二维声波方程的Crank-Nicolson无条件稳定方法研究

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振动与冲击 ›› 2017, Vol. 36 ›› Issue (17) : 79-84.

论文

富志凯，石立华，黄正宇，付尚琛

作者信息 +

A Crank-Nicolson Unconditionally Stable Method for Solving 2D Acoustic Wave Equation

Fu Zhi-Kai Shi Li-Hua Huang Zheng-Yu Fu Shang-Chen

Author information +

文章历史 +

摘要

一阶速度-压力声波方程的有限差分数值模拟中，由于受到Courant-Friedrich-Levy (CFL)稳定性条件的限制，在分析精细结构问题时往往效率低下。本文将Crank-Nicolson（CN）方法引入到声波方程的有限差分模拟中，给出了声波方程的CN差分格式。通过Von Neuman 方法推导分析了CN方法的稳定性条件，证明了该方法的无条件稳定性。同时，采用非均匀网格技术进行网格剖分，进一步提高了仿真效率，减少了内存消耗。仿真实验中，建立了二维多层精细结构的声传播模型，通过与传统时域有限差分的仿真结果进行对比分析，验证了本方法的有效性。

Abstract

Considering the restriction of the Courant-Friedrich-Levy (CFL) stability condition, it is time-consuming to solve the one-order velocity-pressure acoustic wave equation by conventional finite-difference time domain (FDTD) method, especially in analyzing the fine structure problems. This paper introduces the Crank-Nicolson (CN) method to solve the acoustic wave equation. Based on Von Neuman method, the unconditional stability of the CN method of the acoustic wave equation is proved. With the proposed method, the time step will not be restricted by the CFL stability condition any more. Meanwhile, the non-uniform grid technology is used to generate the mesh grid, which further saves the CPU memory and improves the efficiency. In the simulation, a 2-dimentional multi-layers model with fine structure is established. By comparing the simulation results between the traditional FDTD method and the CN method, the effectiveness of the proposed method is proved.

导出引用

富志凯，石立华,黄正宇，付尚琛. 二维声波方程的Crank-Nicolson无条件稳定方法研究[J]. 振动与冲击, 2017, 36(17): 79-84

Fu Zhi-Kai Shi Li-Hua Huang Zheng-Yu Fu Shang-Chen . A Crank-Nicolson Unconditionally Stable Method for Solving 2D Acoustic Wave Equation[J]. Journal of Vibration and Shock, 2017, 36(17): 79-84

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