Numerical solution of partial differential equations on parallel computers using domain decomposition usually requires synchronization and communication among the processors. These operations often have a significant overhead in terms of time and energy. In this paper, we propose communication-efficient parallel algorithms for solving partial differential equations that alleviate this overhead. First, we describe an asynchronous algorithm that removes the requirement of synchronization and checks for termination in a distributed fashion while maintaining the provision to restart iterations if necessary. Then, we build on the asynchronous algorithm to propose an event-triggered communication algorithm that communicates the boundary values to neighboring processors only at certain iterations, thereby reducing the number of messages while maintaining similar accuracy of solution. We demonstrate our algorithms on a successive over-relaxation solver for the Pressure Poisson equation arising from variable density incompressible multiphase flows in 3-D and show that our algorithms improve time and energy efficiency.

翻译：在平行计算机上使用域分解的局部差分方程式的计算式解决方案通常需要处理器之间的同步和沟通。 这些操作通常在时间和能量方面有相当大的间接成本。 在本文中,我们提出了解决部分差分方程式的通信高效平行算法,以缓解这种间接成本。 首先,我们描述了一种非同步的算法,以分散的方式消除同步和检查终止的要求,同时在必要时保留恢复迭代的规定。 然后,我们利用不同步的算法,提出一种事件错开的通信算法,将边界值仅在某些迭代点传递给邻近处理器,从而减少电文的数量,同时保持类似的解决方案的准确性。 我们用一个连续的超宽松解算法,用一种因3D的压强多相位流的可变密度而产生的压力波森方程式的超宽松解算法,并表明我们的算法提高了时间和能源效率。