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基于声波方程扩充的哈密尔顿系统,本文给出了空间精度为八阶的近似解析离散化(NAD)保辛分部Runge-Kutta方法,简称八阶NSPRK方法。该方法采用八阶精度的近似解析离散算子近似空间高阶偏微分算子,并使用二阶精度的辛分部Runge-Kutta方法进行时间离散。我们从理论和数值计算两个方面研究了八阶NSPRK方法的稳定性条件和数值频散关系,并同四阶NSPRK方法、八阶Lax-Wendroff(LWC)方法和八阶交错网格(SG)方法进行了比较。结果表明八阶NSPRK方法压制数值频散的能力显著优于传统数值计算方法。与四阶NSPRK方法和传统四阶辛格式(SPRK)方法相比,八阶NSPRK方法具有最小的数值误差和最高的计算效率:在达到同样消除数值频散的前提下,八阶NSPRK方法的计算速度约为四阶NSPRK方法的2.5倍、为四阶SPRK方法的3.4倍;八阶NSPRK方法的存储量仅为四阶NSPRK方法的47.17%、为四阶SPRK方法的49.41%。在双层介质、非均匀介质和Marmousi等复杂速度模型中,八阶NSPRK方法模拟得到的波场快照非常清晰,无可见数值频散。这些结果表明,八阶NSPRK方法在粗网格条件下能有效地压制数值频散,从而能够极大地节省计算内存,提高计算速度。总体而言,八阶NSPRK方法是一种在地震探测领域和地震学研究中有着巨大应用潜力的数值计算方法。
Based on the Hamiltonian system expanded by acoustic wave equation, this paper presents the Runge-Kutta method of approximate analytic discretization (NAD) and the eighth-order NSPRK method with space precision of eight order. This method uses the eighth-order precision to approximate the higher-order partial differential operators in the approximation space of discrete operators and uses the second-order precision symplectic Runge-Kutta method to perform the time discretization. We study the stability condition and numerical dispersion relation of the eighth-order NSPRK method from two aspects of theory and numerical calculation. The fourth-order NSPRK method, the eighth-order Lax-Wendroff (LWC) method and the eighth order staggered grid (SG) The method is compared. The results show that the ability of the eighth-order NSPRK method to suppress numerical dispersion is significantly better than the traditional numerical method. Compared with the fourth-order NSPRK method and the traditional fourth-order symplectic method (SPRK), the eighth-order NSPRK method has the smallest numerical error and the highest computational efficiency. The computation of the eighth-order NSPRK method achieves the same elimination of numerical dispersion The speed is about 2.5 times that of the fourth-order NSPRK method and 3.4 times that of the fourth-order SPRK method. The storage of the eighth-order NSPRK method is only 47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method. In complex velocity models such as double-layer media, inhomogeneous media and Marmousi, the snapshots of wavefields simulated by the eighth-order NSPRK method are very clear and have no visible numerical dispersion. These results show that the eighth-order NSPRK method can effectively suppress the numerical dispersion under the condition of coarse grid, which can greatly reduce the computational memory and increase the computational speed. In general, the eighth-order NSPRK method is a numerical method that has great potential in seismic exploration and seismology.