Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional MPI processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time-to-solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iterations count when applied to a nonlinear Navier-Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping.
翻译:在模拟基于时间的数字 PDE 中,平行时间方法越来越受欢迎,允许在空间平行饱和时高效使用额外的 MPI 进程。 多数方法在空间和时间上分别处理解决方案和平行问题。 相反, 全自动方法直接解决全时时间系统, 基本上将时间作为另一个空间层面处理。 全自动方法为分别处理空间和时间提供了一些好处, 最显著的是, 显著地增加平行和更快的时间到溶( 在适用的情况下) 。 然而, 快速、 可缩放的全在线方法的开发主要局限于基于时间的( 向上) 应用的( 向上) 快速、 可缩放的( 向上) 进程。 与此相反, 全自动方法直接解决了全时系统, 基本上把众所周知的空间阻断概念作为空间时间和时间的前提条件, 我们开发了一个块性先决条件, 其应用需要空间- 时间( 向上) 最小空间- 递增速度方程方程式的方位方程方程方程, 与压- 压- 平流中每个空间- 的平流压- 的平流的平流法 都显示一个独立空间- 的平流的平时压- 的平时压- 的平流 的平流 的平时压- 度 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度- 度-