Abstract In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the alternating direction implicit (ADI) method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on graphics processing unit (GPU), GPU + Open Multi-Processing (OpenMP), and Hybrid (GPU + OpenMP + message passing interface (MPI)) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, Compute Unified Device Architecture (CUDA), and CUDA + OpenMP implementations.

中文翻译：

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二维声波方程的并行混合实现

摘要 在本文中，我们提出了一种混合并行编程方法，用于使用单台计算机的隐式差分格式的二维声波方程的数值解。计算是在隐式有限差分方案中进行的。首先，我们将微分方程转化为隐式有限差分方程，然后使用交替方向隐式（ADI）方法，我们将方程分解为两个子方程。使用循环归约算法，我们计算了一个近似解。最后，我们将此算法更改为在图形处理单元 (GPU)、GPU + 开放式多处理 (OpenMP) 和混合 (GPU + OpenMP + 消息传递接口 (MPI)) 计算平台上并行化。特别关注的是提高并行算法的性能，以根据执行时间计算加速度。我们通过将我们的结果与在经典处理器内核、统一计算设备架构 (CUDA) 和 CUDA + OpenMP 实现上运行相同模拟所获得的结果进行比较，表明在混合方法上运行的代码给出了预期结果。