We have developed compact implicit finite-difference (FD) schemes in the time-space domain based on the second-order FD approximation for accurate solution of the acoustic wave equation in 1D, 2D, and 3D. Our method is based on the weighted linear combination of the second-order FD operators with different spatial orientations to mitigate numerical error anisotropy and the weighted averaging of the mass acceleration term over the grid points of the second-order FD stencil to reduce the overall numerical dispersion error. We have developed a derivation of the schemes for 1D, 2D, and 3D cases. We obtain their corresponding dispersion equations, then we find the optimum weights by optimization of the time-space domain dispersion function, and finally we tabulate the optimized weights for each case. We analyze the numerical dispersion, stability, and convergence rates of our schemes and compare their numerical dispersion characteristics with the standard high-order ones. We also discuss the efficient solution of the system of equations associated with our implicit schemes using the conjugate-gradient method. The comparison of dispersion curves and the numerical solutions with the analytical and the pseudospectral solutions reveals that our schemes have better performance than the standard spatial high-order schemes and remain stable for relatively large time steps.

中文翻译：

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时空域声波方程的二阶隐式有限差分格式

我们已经开发了基于二阶 FD 近似的时空域紧凑隐式有限差分 (FD) 方案，用于精确求解 1D、2D 和 3D 中的声波方程。我们的方法基于具有不同空间方向的二阶 FD 算子的加权线性组合，以减轻数值误差各向异性，并基于二阶 FD 模板网格点上的质量加速度项的加权平均，以减少整体数值色散误差。我们已经为 1D、2D 和 3D 案例开发了方案的推导。我们得到它们对应的色散方程，然后通过优化时空域色散函数找到最优权重，最后我们将每种情况下的优化权重制成表格。我们分析了数值色散、稳定性、和我们方案的收敛速度，并将它们的数值色散特性与标准的高阶特性进行比较。我们还使用共轭梯度方法讨论了与我们的隐式方案相关联的方程组的有效解。色散曲线和数值解与解析解和伪谱解的比较表明，我们的方案比标准空间高阶方案具有更好的性能，并且在相对较大的时间步长内保持稳定。