Abstract This work studies the parallelization and empirical convergence of two finite difference acoustic wave propagation methods on 2-D rectangular grids, that use the same alternating direction implicit (ADI) time integration. This ADI integration is based on a second-order implicit Crank-Nicolson temporal discretization that is factored out by a Peaceman-Rachford decomposition of the time and space equation terms. In space, these methods highly diverge and apply different fourth-order accurate differentiation techniques. The first method uses compact finite differences (CFD) on nodal meshes that requires solving tridiagonal linear systems along each grid line, while the second one employs staggered-grid mimetic finite differences (MFD). For each method, we implement three parallel versions: (i) a multithreaded code in Octave, (ii) a C++ code that exploits OpenMP loop parallelization, and (iii) a CUDA kernel for a NVIDIA GTX 960 Maxwell card. In these implementations, the main source of parallelism is the simultaneous ADI updating of each wave field matrix, either column-wise or row-wise, according to the differentiation direction. In our numerical applications, the highest performances are displayed by the CFD and MFD CUDA codes that achieve speedups of 7.21x and 15.81x, respectively, relative to their C++ sequential counterparts with optimal compilation flags. Our test cases also allow to assess the numerical convergence and accuracy of both methods. In a problem with exact harmonic solution, both methods exhibit convergence rates close to 4 and the MDF accuracy is practically higher. Alternatively, both convergences decay to second order on smooth problems with severe gradients at boundaries, and the MDF rates degrade in highly-resolved grids leading to larger inaccuracies. This transition of empirical convergences agrees with the nominal truncation errors in space and time.

中文翻译：

---

有限差分声波传播的交替方向隐式时间积分：并行化和收敛

摘要 这项工作研究了两种有限差分声波传播方法在二维矩形网格上的并行化和经验收敛，这些方法使用相同的交替方向隐式 (ADI) 时间积分。此 ADI 积分基于二阶隐式 Crank-Nicolson 时间离散化，该离散化由时间和空间方程项的 Peaceman-Rachford 分解分解。在空间中，这些方法高度发散并应用不同的四阶精确微分技术。第一种方法在节点网格上使用紧凑有限差分 (CFD)，需要沿每条网格线求解三对角线性系统，而第二种方法采用交错网格模拟有限差分 (MFD)。对于每种方法，我们实现了三个并行版本：(i) Octave 中的多线程代码，(ii) 利用 OpenMP 循环并行化的 C++ 代码，以及 (iii) 用于 NVIDIA GTX 960 Maxwell 卡的 CUDA 内核。在这些实现中，并行性的主要来源是每个波场矩阵的同步 ADI 更新，无论是按列还是按行，根据微分方向。在我们的数值应用程序中，最高性能由 CFD 和 MFD CUDA 代码显示，相对于具有最佳编译标志的 C++ 顺序对应代码，它们分别实现了 7.21 倍和 15.81 倍的加速。我们的测试用例还允许评估两种方法的数值收敛性和准确性。在精确谐波解的问题中，两种方法的收敛速度都接近 4，MDF 精度实际上更高。或者，在边界处具有严重梯度的平滑问题上，两种收敛都衰减到二阶，并且 MDF 速率在高分辨率网格中会降低，从而导致更大的不准确性。这种经验收敛的转变与空间和时间的名义截断误差一致。