Department of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran.
J Acoust Soc Am. 2013 Sep;134(3):1775-90. doi: 10.1121/1.4774278.
Various numerical algorithms have been developed to solve the Khokhlov-Kuznetsov-Zabolotskaya (KZK) parabolic nonlinear wave equation. In this work, a generalized time-domain numerical algorithm is proposed to solve the diffraction term of the KZK equation. This algorithm solves the transverse Laplacian operator of the KZK equation in three-dimensional (3D) Cartesian coordinates using a finite-difference method based on the five-point implicit backward finite difference and the five-point Crank-Nicolson finite difference discretization techniques. This leads to a more uniform discretization of the Laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term. In addition, a new empirical algorithm based on the LU decomposition technique is proposed to solve the system of linear equations obtained from this discretization. The proposed empirical algorithm improves the calculation speed and memory usage, while the order of computational complexity remains linear in calculation of the diffraction term in the KZK equation. For evaluating the accuracy of the proposed algorithm, two previously published algorithms are used as comparison references: the conventional 2D Texas code and its generalization for 3D geometries. The results show that the accuracy/efficiency performance of the proposed algorithm is comparable with the established time-domain methods.
已经开发出各种数值算法来求解 Khokhlov-Kuznetsov-Zabolotskaya（KZK）抛物线非线性波动方程。在这项工作中，提出了一种广义时域数值算法来求解 KZK 方程的绕射项。该算法在三维笛卡尔坐标系中使用五点隐式向后有限差分和五点 Crank-Nicolson 有限差分离散化技术来求解 KZK 方程的横向拉普拉斯算子。这导致拉普拉斯算子的离散化更加均匀，从而在不影响绕射项准确性的情况下减少了计算网格节点的数量。此外，还提出了一种基于 LU 分解技术的新的经验算法来求解由此离散化得到的线性方程组。所提出的经验算法提高了计算速度和内存使用效率，而计算复杂度的阶数在 KZK 方程的绕射项计算中保持线性。为了评估所提出算法的准确性，使用了两个已发表的算法作为比较参考：传统的 2D Texas 代码及其用于 3D 几何形状的推广。结果表明，所提出算法的准确性/效率性能与已建立的时域方法相当。
